3-5 Polynomials, Linear Factors, and Zeros

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, P(x) = 2x^3 - 4x^2 + 3x - 5 is a polynomial.

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What are linear factors of a polynomial?

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Linear factors are expressions of the form (x - r), where r is a root (zero) of the polynomial. A polynomial can be factored into linear factors if it can be expressed as a product of these factors.

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CCSS.HSA.APR.B.2

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How do you find the zeros of a polynomial?

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for the variable. For example, for P(x) = x^2 - 4, set x^2 - 4 = 0, which gives x = 2 and x = -2.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial. For example, in P(x) = 3x^4 + 2x^3 - x + 7, the degree is 4.

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What is a cubic trinomial?

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A cubic trinomial is a polynomial of degree 3 with three terms. For example, P(x) = 2x^3 + 3x^2 - 5 is a cubic trinomial.

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What is the vertical stretch factor in a polynomial?

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The vertical stretch factor, often denoted as 'a' in the equation y = a * f(x), affects the steepness of the graph. If |a| > 1, the graph stretches vertically; if 0 < |a| < 1, it compresses vertically.

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How do you factor a polynomial?

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To factor a polynomial, look for common factors, use grouping, or apply special factoring techniques like difference of squares or perfect square trinomials. For example, x^2 - 9 can be factored as (x - 3)(x + 3).

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CCSS.HSA.APR.C.4

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What is the relationship between the zeros and the factors of a polynomial?

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The zeros of a polynomial correspond to the values of x that make the polynomial equal to zero. Each zero (r) gives rise to a linear factor (x - r).

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CCSS.HSF-IF.C.7C

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or root) of a polynomial equation with integer coefficients is of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Tags

CCSS.HSA.APR.B.2

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form (x - r). It is faster than long division and is used to find the quotient and remainder.

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CCSS.HSA.APR.D.6