Precalculus by Richard Wright

Anger is cruel and fury overwhelming, but who can stand before jealousy? Proverbs 27:4 NIV

Summary: In this section, you will:

- Evaluate a polynomial using the Remainder Theorem.
- Use the Factor Theorem to solve a polynomial equation.
- Use the Rational Zero Theorem to find rational zeros.

SDA NAD Content Standards (2018): PC.6.2, PC.6.4

A landscape company is going to put some decorative rectangular prism-shaped stepping stones to make a path across a creek. Each stone will use 648 cubic inches of cement because that is convenient based on their cement supply. They decided that having the width six inches greater than the length is a pleasing proportion, and that the height should be one fourth the width for strength. What should be the dimensions of the stepping stone?

This problem can be solved by writing a cubic function and solving a cubic equation for the volume of the stepping stone. This lesson highlights a variety of tools for writing polynomial functions and solving polynomial equations.

The Remainder Theorem provides a convenient way to evaluate polynomials based on division. A polynomial may be evaluated at *f*(*k*) by dividing it by *x* − *k*. Synthetic division makes the process quick. Here is the proof of the theorem.

Remember that the Division Algorithm states that, given a polynomial dividend *f*(*x*) and a non-zero polynomial divisor *d*(*x*) where the degree of *d*(*x*) is less than or equal to the degree of *f*(*x*), there exist unique polynomials *q*(*x*) and *r*(*x*) such that

*f*(*x*) = *d*(*x*) *q*(*x*) + *r*(*x*)

Let the divisor be *d*(*x*) = *x* − *k*. Then the Division Algorithm becomes

*f*(*x*) = (*x* − *k*)q(*x*) + *r*

Since the divisor *x* − *k* is linear, the remainder will be a constant, *r*. And, if *x* = *k*, then

*f*(*k*) = (*k* − *k*)*q*(*k*) + *r*

= 0 · *q*(*k*) + *r*

= *r*

In other words, *f*(*k*) is the remainder obtained by dividing *f*(*x*) by *x* − *k*.

If a polynomial *f*(*x*) is divided by *x* − *k*, then the remainder is the value *f*(*k*).

- Use synthetic division to divide the polynomial by
*x*−*k*. - The remainder is the value
*f*(*k*).

Use the Remainder Theorem to evaluate *f*(*x*) = *x*^{4} − 2*x*^{3} − 3*x*^{2} + *x* − 11 at *x* = 3.

To use the Remainder Theorem, use synthetic division to divide the polynomial by *x* − 3.

$$ \begin{array}{rrrrrr} \underline{3}| & 1 & -2 & -3 & 1 & -11 \\ & & 3 & 3 & 0 & 3\\ \hline & 1 & 1 & 0 & 1 & |\underline{-8} \end{array} $$

The remainder is −8. Therefore, *f*(3) = −8.

**Analysis**

It is possible to check the answer by evaluating *f*(3).

*f*(*x*) = *x*^{4} − 2*x*^{3} − 3*x*^{2} + *x* − 11

*f*(3) = (3)^{4} − 2(3)^{3} − 3(3)^{2} + (3) − 11

= −8

Use the Remainder Theorem to evaluate *f*(*x*) = 3*x*^{5} − *x*^{4} − 2*x*^{3} + *x*^{2} + 3 at *x* = −1.

*f*(−1) = 2

The Factor Theorem tells how the zeros of a polynomial are related to the factors. Remember that the Division Algorithm tells us

*f*(*x*) = (*x* − *k*)*q*(*x*) + *r*.

If *k* is a zero, then the remainder, *r*, is *f*(*k*) = 0 and *f*(*x*) = (*x* − *k*)*q*(*x*) + 0 or *f*(*x*) = (*x* − *k*)*q*(*x*).

Notice, written in this form, *x* − *k* is a factor of *f*(*x*). So, if *k* is a zero of *f*(*x*), then *x* − *k* is a factor of *f*(*x*).

Similarly, if *x* − *k* is a factor of *f*(*x*), then the remainder of the Division Algorithm *f*(*x*) = (*x* − *k*)*q*(*x*) + *r* is 0. This tells that *k* is a zero.

This pair of statements is the Factor Theorem. It turns out that a polynomial of degree *n* in the complex number system will have *n* zeros. The Factor Theorem can be used to completely factor a polynomial into the product of *n* factors. Once the polynomial has been completely factored, its zeros can easily be found using the Zero Product Property.

According to the Factor Theorem, *k* is a zero of *f*(*x*) if and only if (*x* − *k*) is a factor of *f*(*x*).

- Use synthetic division to divide the polynomial by the given factor, (
*x*−*k*). - Confirm that the remainder is 0.
- If the quotient is NOT a quadratic, repeat steps 1 and 2 with another factor using the quotient as the polynomial.
- If the quotient IS a quadratic, factor the quadratic quotient if possible.
- Write the polynomial as the product of factors.

Show that (*x* + 1) is a factor of *x*^{3} + 2*x*^{2} − 5*x* − 6. Find the remaining factors. Use the factors to determine the zeros of the polynomial.

Use synthetic division to show that (*x* + 1) is a factor of the polynomial.

$$ \begin{array}{rrrrr} \underline{-1}| & 1 & 2 & -5 & -6 \\ & & -1 & -1 & 6 \\ \hline & 1 & 1 & -6 & |\underline{\phantom{0}0} \end{array} $$

The remainder is zero, so (*x* + 1) is a factor of the polynomial. The quotient is *x*^{2} + *x* − 6 which is a quadratic. Factor that quadratic.

*x*^{2} + *x* − 6 = (*x* − 2)(*x* + 3)

Now write all the factors of the the polynomial as

(*x* + 1)(*x* − 2)(*x* + 3)

The Factor Theorem says that if *x* − *k* is a factor then *k* is a zero of the polynomial. Thus, the zeros of *x*^{3} + 2*x*^{2} − 5*x* − 6 are –1, 2, and −3.

Use the Factor Theorem to find the zeros of *f*(*x*) = *x*^{3} − 5*x*^{2} − 10*x* + 24 given that (*x* − 4) is a factor of the polynomial.

The zeros are −2, 3, and 4.

Another use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first there needs to be a group of rational numbers to test. There are an infinite number of possible zeros to choose from. It would be nice to have fewer numbers to choose from. The Rational Zero Theorem narrows down the number of possible rational zeros using the ratio of the factors of the constant term and factors of the leading coefficient of the polynomial.

Think about a quadratic function with two zeros, \(x = \frac{2}{3}\) and \(x = \frac{4}{5}\). The Factor Theorem indicates that if a zero is *k*, then a factor is (*x* − *k*). If *k* is a fraction, then the factor can be found by setting *x* = *k*.

$$ x = \frac{2}{3}; x = \frac{4}{5} $$

Subtract to make these equal zero.

$$ x - \frac{2}{3} = 0; x - \frac{4}{5} = 0 $$

Multiply by the denominators to get rid of the fraction.

$$ 3x - 2 = 0; 5x - 4 = 0 $$

Multiply these together to make the quadratic function.

$$ \left(3x - 2\right)\left(5x - 4\right) $$

Multiply these together.

$$ 15x^2 - 22x + 8 $$

Notice that the leading coefficient and constant term can be factored.

$$ (3·5)x^2 - 22x + (2·4) $$

The zeros are made of ratios of the factors of the constant term to the factors of the leading coefficient: \(\frac{2}{3}\) and \(\frac{4}{5}\). This is true of all polynomials and is called the Rational Zero Theorem.

If the polynomial \(f(x) = a_{n}x^n + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_0\) has integer coefficients, then every rational zero *f*(*x*) has the form of \(\frac{p}{q}\) where *p* is a factor of the constant term *a*_{0} and *q* is a factor of the leading coefficient *a*_{n}.

When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.

- List all factors of the constant term and all factors of the leading coefficient.
- List all possible values of \(\frac{p}{q}\) where
*p*is a factor of the constant term and*q*is a factor of the leading coefficient. Be sure to include both positive and negative numbers. - Determine which of the possible zeros are actual zeros by using synthetic division. A graph can be used to choose the possible zero by looking for the
*x*-intercepts. - After the first zero is found, use the quotient, or depressed polynomial, to find the next zero.
- Repeat until the depressed polynomial is quadratic, then factor or use the quadratic formula to find the last two zeros.

The quotient that results from dividing a polynomial by a factor is called a depressed polynomial because it is one degree less than the original polynomial.

List all possible rational zeros of *f*(*x*) = 2*x*^{4} + 3*x*^{3} − 5*x*^{2} − 4.

The only possible rational zeros of *f*(*x*) are the ratios of the factors of the last term, –4, and the factors of the leading coefficient, 2.

Find the *p*'s which are factors of the constant term, −4.

*p* = ±1, ±2, ±4.

Find the *q*'s which are factors of the leading coefficient, 2.

*q* = ±1, ±2.

The rational zeros are the ratios of *p* to *q*.

$$ \frac{p}{q} = ±\frac{1}{1}, ±\frac{1}{2}, ±\frac{2}{1}, ±\frac{2}{2}, ±\frac{4}{1}, ±\frac{4}{2} $$

Notice that \(\frac{1}{1} = 1\) and \(\frac{4}{2} = 2\), which have already been listed. So the list can be shortened.

$$ \frac{p}{q} = ±1, ±\frac{1}{2}, ±2, ±4 $$

Use the Rational Zero Theorem to find the rational zeros of *f*(*x*) = 4*x*^{3} + 8*x*^{2} − 31*x* + 4.

**Solution**

The Rational Zero Theorem say that the only possible rational zeros of *f*(*x*) are the ratios of the factors of the last term, 4, and the factors of the leading coefficient, 4.

Find the *p*'s which are factors of the constant term, 4.

*p* = ±1, ±2, ±4.

Find the *q*'s which are factors of the leading coefficient, 4.

*q* = ±1, ±2, ±4.

The rational zeros are the ratios of *p* to *q*.

$$ \frac{p}{q} = ±\frac{1}{1}, ±\frac{1}{2}, ±\frac{1}{4}, ±\frac{2}{1}, ±\frac{2}{2}, ±\frac{2}{4}, ±\frac{4}{1}, ±\frac{4}{2}, ±\frac{4}{4} $$

Simplifying and removing duplicates shortens the list to

$$ \frac{p}{q} = ±1, ±\frac{1}{2}, ±\frac{1}{4}, ±2, ±4 $$

Choose one of the possible rational zeros to test. The graph in figure 2 indicates that −4 would be a good choice because that is a *x*-intercept and *x*-intercepts are zeros. Start with *x* = −4.

$$ \begin{array}{rrrrr} \underline{-4}| & 4 & 8 & -31 & 4 \\ & & -16 & 32 & -4 \\ \hline & 4 & -8 & 1 & |\underline{\phantom{0}0} \end{array} $$

The original function was degree 3. After dividing, the depressed polynomial is degree 2. The depressed polynomial is 4*x*^{2} − 8*x* + 1. Since this is quadratic, factor or use the quadratic formula to find the last two zeros. In this case, the quadratic does not factor, so use the quadratic formula.

$$ x = \frac{-b±\sqrt{b^2 - 4ac}}{2a} $$

$$ x = \frac{8 ± \sqrt{(-8)^2 - 4(4)(1)}}{2(4)} $$

$$ x = \frac{8 ± 4\sqrt{3}}{8} $$

$$ x = \frac{2 ± \sqrt{3}}{2} $$

The zeros are −4, \(\frac{2 + \sqrt{3}}{2}\), and \(\frac{2 - \sqrt{3}}{2}\). Notice the last two zeros are irrational, so the only rational zero is −4.

Use the Rational Zero Theorem to find the rational zeros of *f*(*x*) = *x*^{3} + 4*x*^{2} − 11*x* − 30.

−5, −2, 3

If a polynomial *f*(*x*) is divided by *x* − *k*, then the remainder is the value *f*(*k*).

- Use synthetic division to divide the polynomial by
*x*−*k*. - The remainder is the value
*f*(*k*).

According to the Factor Theorem, *k* is a zero of *f*(*x*) if and only if (*x* − *k*) is a factor of *f*(*x*).

- Use synthetic division to divide the polynomial by the given factor, (
*x*−*k*). - Confirm that the remainder is 0.
- If the quotient is NOT a quadratic, repeat steps 1 and 2 with another factor using the quotient as the polynomial.
- If the quotient IS a quadratic, factor the quadratic quotient if possible.
- Write the polynomial as the product of factors.

If the polynomial \(f(x) = a_{n}x^n + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_0\) has integer coefficients, then every rational zero *f*(*x*) has the form of \(\frac{p}{q}\) where *p* is a factor of the constant term *a*_{0} and *q* is a factor of the leading coefficient *a*_{n}.

When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.

- List all factors of the constant term and all factors of the leading coefficient.
- List all possible values of \(\frac{p}{q}\) where
*p*is a factor of the constant term and*q*is a factor of the leading coefficient. Be sure to include both positive and negative numbers. - Determine which of the possible zeros are actual zeros by using synthetic division. A graph can be used to choose the possible zero by looking for the
*x*-intercepts. - After the first zero is found, use the quotient, or depressed polynomial, to find the next zero.
- Repeat until the depressed polynomial is quadratic, then factor or use the quadratic formula to find the last two zeros.

Helpful videos about this lesson.

- What is the difference between rational zeros and real zeros?
- (2
*x*^{3}+ 5*x*^{2}− 2*x*+ 6) ÷ (*x*− 2) - (
*x*^{4}+*x*+ 1) ÷ (*x*+ 3) *f*(*x*) =*x*^{3}- 4*x*^{2}+*x*+ 6;*x*− 3*f*(*x*) = 2*x*^{3}− 7*x*^{2}− 5*x*+ 4;*x*+ 1*f*(*x*) =*x*^{3}+ 2*x*^{2}− 3*x*− 6;*x*+ 2*f*(*x*) = 2*x*^{3}+*x*^{2}− 12*x*+ 9;*x*+ 3*f*(*x*) = 6*x*^{3}+ 25*x*^{2}+ 21*x*− 10; 2*x*+ 5*f*(*x*) = 2*x*^{3}− 10*x*^{2}− 2*x*+ 7*f*(*x*) = 8*x*^{4}− 5*x*^{2}− 1*x*+ 4- Find the dimensions of the box where the length is four inches greater than the width. The height is three times the width. The volume is 675 cubic inches.
- (2-04) Using synthetic division, decide if the first expression is a factor of the second: (
*x*+ 3), (2*x*^{3}+ 7*x*^{2}+ 4*x*+ 3). - (2-04) Divide using long division: (2
*x*^{3}+ 7*x*^{2}+ 4*x*+ 3) ÷ (*x*+ 3). - (2-03) Find the zeros and multiplicity of each zero for
*h*(*x*) =*x*^{4}− 6*x*^{3}+ 9*x*^{2}. - (2-02) Find the minimum of
*x*^{2}− 4*x*+ 6. - (2-01) Simplify \(\frac{2+\sqrt{-27}}{1-\sqrt{-9}}\).
- (1-10) Find the equation of the best-fitting line for the following table:

*x*1 3 5 7 *y*1.5 0.5 −0.5 −1.5 - (1-05) Find the zeros of
*f*(*x*) = −3*x*+ 6. - (1-04) Given the function
*g*(*x*) = 2*x*^{2}, evaluate \(\frac{g(x+h)-g(x)}{h}\). - (1-03) Find the equation of the line perpendicular to
*y*= −3*x*and passing through the point (2, 4).

Use the Remainder Theorem to find the remainder.

Use the Factor Theorem to find all real zeros for the given polynomial function and one factor.

List all possible rational zeros for the functions.

Problem Solving

Mixed Review

- Rational zeros can be written as fractions, but real zeros include irrational numbers which cannot be written as fractions.
- 38
- 79
- −1, 2, 3
- −1, \(\frac{1}{2}\), 4
- −2, \(-\sqrt{3}\), \(\sqrt{3}\)
- −3, 1 \(\frac{3}{2}\)
- \(-\frac{5}{2}\), −2, \(\frac{1}{3}\)
- \(±1, ±\frac{1}{2}, ±7, ±\frac{7}{2}\)
- \(±1, ±\frac{1}{2}, ±\frac{1}{4}, ±\frac{1}{8}, ±2, ±4\)
- 9 × 5 × 15 inches
- Yes
- 2
*x*^{2}+*x*+ 1 - 0 with multiplicity 2, 3 with multiplicity 2
- (2, 2)
- \(\frac{2-9\sqrt{3}}{10}+\frac{6+3\sqrt{3}}{10}i\)
*y*= −0.5*x*+ 2- 2
- 4
*x*+ 2*h* - \(y = \frac{1}{3}x + \frac{10}{3}\)