Equivalent Representations of Polynomial Expressions (College Board AP® Precalculus): Revision Note

Equivalent representations of polynomial expressions

What are the different forms of a polynomial expression?

• A polynomial function can be written in different equivalent forms

  • each of which reveals different information about the function

• The two most important forms for polynomial functions are standard form and factored form

What is standard form?

• The standard form of a polynomial is the expanded form written as a sum of terms in descending order of degree

  • $p\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\dots +a_2{x}^2+a_{1}x+a_0$

    • where $a_n\ne 0$

  • This is the same form introduced in the Polynomial Functions study guide

• Standard form makes it easy to see

  • The degree of the polynomial (the highest power of $x$)

  • The leading term ($a_n{x}^n$) and the leading coefficient ($a_n$)

  • The end behavior of the function (determined by the degree and sign of the leading coefficient, as covered in the End Behavior of Polynomial Functions study guide)

  • The constant term ($a_0$), which gives the $\mathit{y}$-intercept of the graph (since $p\left(0\right)=a_0$)

    • so the $y$-intercept is at $\left(0,a_0\right)$

What is factored form?

• The factored form of a polynomial expresses the function as a product of linear factors

  • E.g. $p\left(x\right)=-2(x-1)(x+3)(x-5)$

  • There may also be an irreducible quadratic factor if there are non-real zeros

    • E.g. $p\left(x\right)=3\left(x-2\right)\left(x^2+2x+5\right)$

• Factored form makes it easy to see

  • The real zeros of the function (the values of $x$ that make each factor equal to zero)

  • The multiplicity of each zero (from repeated factors)

  • The $\mathit{x}$-intercepts of the graph (since the real zeros correspond to points where the graph crosses or touches the $x$-axis)

  • E.g. from $p\left(x\right)=-2(x-1)(x+3)(x-5)$

    • The zeros are $x=1$, $x=-3$, and $x=5$ (each with multiplicity 1)

    • The $x$-intercepts are $(1, 0)$, $(-3, 0)$, and $(5, 0)$

How do I convert between forms?

• From factored form to standard form

  • Expand (multiply out) the factors and collect like terms

  • E.g. $p\left(x\right)=(x-2)(x+5)=x^2+5x-2x-10=x^2+3x-10$

• From standard form to factored form

  • Factor the polynomial expression

  • For this course, factoring without technology is limited to:

    • Factoring out a common factor

      • E.g. $3x^3-6x^2=3x^2(x-2)$

    • Factoring quadratics

      • E.g. $x^2+3x-10=(x+5)(x-2)$

  • For higher-degree polynomials, you may need to combine these techniques

    • E.g. $p\left(x\right)=x^3+2x^2-15x=x(x^2+2x-15)=x(x+5)(x-3)$

    • First factor out the common factor $x$, then factor the remaining quadratic

Why is it useful to have different forms?

• Different forms make different information accessible

• For example

  • If you need to determine end behavior, use standard form

    • the degree and leading coefficient are immediately visible

  • If you need to find zeros or $\mathit{x}$-intercepts, use factored form

    • the zeros can be read directly from the factors

  • If you need to find the $\mathit{y}$-intercept, either form works

    • but standard form makes it easiest (the constant term $a_0$)

• Being able to convert between forms means you can extract whatever information is needed from a single polynomial function