introduction synthetic division, along with your knowledge of end behavior and turning points, can...

IntroductionSynthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows for more accurate sketches of functions.

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2.3.3: Finding Zeros

Key Concepts• Recall that roots are the x-intercepts of a function. In other

words, these are the x-values for which a function equals 0. These zeros are another way of referring to the roots of a function.

• When a polynomial equation with a degree greater than 0 is solved, it may have one or more real solutions, or it may have no real solutions (in which case it would have complex solutions).

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Key Concepts• Recall that both real and imaginary numbers belong to the

set of complex numbers; therefore, all polynomial functions with a degree greater than 0 will have at least one root in the set of complex numbers. This is referred to as the Fundamental Theorem of Algebra.

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Fundamental Theorem of Algebra

If p(x) is a polynomial function of degree n ≥ 1 with complex coefficients, then the related equation p(x) = 0 has at least one complex solution (root).

Key Concepts, continued• A repeated root is a root that occurs more than once

in a polynomial function.

• Recall that the solutions to a quadratic equation that contains imaginary numbers come in pairs. These are called complex conjugates, the complex number that when multiplied by another complex number produces a value that is wholly real; the complex conjugate of a + bi is a – bi.

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Key Concepts, continued• If an imaginary number is a zero of a function, its

conjugate is also a zero of that function. This is true for all polynomial functions, and is known as the Complex Conjugate Theorem.

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Complex Conjugate Theorem

Let p(x) be a polynomial with real coefficients. If a + bi is a root of the equation p(x) = 0, where a and b are real and b ≠ 0, then a – bi is also a root of the equation.

Key Concepts, continued• For a polynomial function p(x), the factor of a

polynomial is any polynomial that divides evenly into that function. Recall that when a polynomial is divided by one of its factors, there is a remainder of 0 and the result is a depressed polynomial. This is an illustration of the Factor Theorem.

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Factor Theorem

The binomial x – a is a factor of the polynomial p(x) if and only if p(a) = 0, where a is a real number.

Key Concepts, continued• The Factor Theorem can help to find all the factors of a

polynomial. • To do this, first show that the binomial in question is a

factor of the polynomial. If the remainder is 0, then the binomial is a factor.

• Then, determine if the resulting depressed polynomial can be factored.

• The identified factors indicate where the function crosses the x-axis.

• The zeros of a function are related to the factors of the polynomial. The graph of a polynomial function shows the zeros of the function, which are the x-intercepts of the graph. 7


Key Concepts, continued• It is often helpful to know which integer values of a to

try when determining p(a) = 0.

• Use the Integral Zero Theorem to determine the zeros of a polynomial function.

• Identify the factors of the constant term of a polynomial function and use substitution to determine if each number results in a zero.

• Use synthetic division to determine the remaining factors.

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Key Concepts, continued

• For example, consider the equation p(x) = x2 + 10x + 25. an = 1 and a0 = 25. A possible zero of this function is –5

because –5 is a factor of 25.

• The zeros of any polynomial function correspond to the x-intercepts of the graph and to the roots of the corresponding equation. 9


Integral Zero Theorem

If the coefficients of a polynomial function are integers such that an = 1 and a0 ≠ 0, then any rational zeros of the function must be factors of a0.

Key Concepts, continued• If a polynomial has a factor x – a that is repeated n

times, then the root is called a repeated root and x = a is a zero of multiplicity. Multiplicity refers to the number of times a zero of a polynomial function occurs.

• If the multiplicity is odd, then the graph intersects the x-axis at the point (x, 0). If the multiplicity is even, then the graph just touches the axis at the point (x, 0).

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Key Concepts, continued• If p(x) is a polynomial with real coefficients whose terms

are arranged in descending powers of the variable, then the number of positive zeros of y = p(x) is the same as the number of sign changes of the coefficients of the terms, or is less than this by an even number. Recall that when solving a quadratic function, we used the quadratic formula, which generated pairs of roots. Often, these roots were complex and the x-intercepts could not be graphed on the coordinate plane. For this reason, we must count down from our maximum number of zeros by twos to determine the number of real zeros. For example, for a polynomial with 3 sign changes, the number of positive zeros may be 3 or 1, since 3 – 2 = 1. 11


Key Concepts, continued• Also, the number of negative zeros of y = p(x) is the

same as the number of sign changes of the coefficients of the terms of p(–x), or is less than this number by an even number.

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Common Errors/Misconceptions• not finding all the factors of a polynomial function • making sign errors when performing synthetic division • misusing the terms “roots” and “factors”

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Guided Practice

Example 1Given the equation x3 + 4x2 – 3x – 18 = 0, state the number and type of roots of the equation if one root is –3.

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Guided Practice: Example 1, continued

1. Use synthetic division to find the depressed polynomial.The related polynomial is x3 + 4x2 – 3x – 18, with coefficients 1, 4, –3, and –18.

One root of the equation is –3; therefore, a factor of the related polynomial is [x – (–3)], which simplifies to x + 3.

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Guided Practice: Example 1, continuedDivide the related polynomial by x + 3 to find the depressed polynomial. Let the value of a be –3.

The depressed polynomial is x2 + x – 6.

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2. Factor the depressed polynomial to find the remaining factors.Use previously learned strategies to factor the polynomial.

x2 + x – 6 Depressed polynomial

(x – 2)(x + 3) Factor the polynomial.

The remaining factors are (x – 2) and (x + 3).

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3. State the number of roots and type of roots of the equation.The factors of the related polynomial are (x + 3), (x – 2), and (x + 3). Recall that we divided the depressed polynomial by (x + 3) before finding the remaining factors, so we must include (x + 3) as a factor. Therefore, the roots of the equation are –3, –3, and 2. Since –3 appears twice, it is a repeated root. Because of the repeated root, this equation has only 2 real roots: –3 and 2.

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Guided Practice

Example 3Write the simplest polynomial function with integral coefficients that has the zeros 5 and 3 – i.

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1. Determine additional zeros of the function.Because 3 – i is a zero, then according to the Complex Conjugate Theorem, the conjugate 3 + i is also a zero.

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2. Use the zeros to write the polynomial as a product of the factors.The zeros are 5, 3 – i, and 3 + i. These can be written as the factors x – 5, x – (3 – i), and x – (3 + i).

The polynomial function written in factored form with zeros 5 and 3 – i is f(x) = (x – 5)[x – (3 – i )][x – (3 + i )].

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3. Multiply the factors to determine the polynomial function.

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f(x) = (x – 5)[x – (3 – i )][x – (3 + i )] Polynomial function written in factored form

f(x) = (x – 5)[(x – 3) – i ][(x – 3) + i ] Regroup terms.

f(x) = (x – 5)[(x – 3)2 – i 2]

Rewrite as the difference of two squares.

f(x) = (x – 5)(x2 – 6x + 9 – i 2) Simplify.


The polynomial function of least degree with integral coefficients whose zeros are 5 and 3 – i is f(x) = x3 – 11x2 + 40x – 50.

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f(x) = (x – 5)[x2 – 6x + 9 – (–1)]Replace i

2 with –1, since i

2 = –1.

f(x) = (x – 5)(x2 – 6x + 9 + 1) Simplify.

f(x) = x3 – 11x2 + 40x – 50 Distribute.

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introduction synthetic division, along with your knowledge of end behavior and turning points, can...

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