06/14 6. Polynomial Expressions and Functions Free Online FlashCards

What exponents are allowed in a polynomial term $ax^n$?

The exponent $n$ must be a nonnegative integer, such as $0,1,2,3$, and so on.

What is the degree of $f(x)=4x^5-3x^2+7$?

The degree is $5$, because $5$ is the largest exponent with a nonzero coefficient.

How are polynomials added or subtracted?

Combine like terms: terms with the same variable raised to the same power.

How are two polynomials multiplied?

Use the distributive property: multiply every term in one polynomial by every term in the other, then combine like terms.

How can $9x^2-25$ be factored?

Apply the difference-of-squares identity: $a^2-b^2=(a-b)(a+b)$. Thus, $9x^2-25=(3x-5)(3x+5)$.

What does the Polynomial Division Algorithm state?

$f(x)=d(x)q(x)+r(x)$, where the degree of $r(x)$ is less than the degree of the nonzero divisor $d(x)$.

Divide $2x^3+3x^2-5x+6$ by $x+2$.

The quotient is $2x^2-x-3$, and the remainder is $12$.

When is synthetic division used?

Synthetic division is a shortened method for division by a linear polynomial $x-c$; it uses only the dividend's coefficients.

What is the remainder when $x^3-4x^2+2x+7$ is divided by $x-3$?

By the Remainder Theorem, the remainder is $f(3)=4$.

State the Factor Theorem.

$x-c$ is a factor of $f(x)$ if and only if $f(c)=0$.

What is a zero of a polynomial function?

A zero is an input $c$ for which $f(c)=0$; it is also the $x$-coordinate of an $x$-intercept.

What are the zero multiplicities of $(x-2)^3(x+1)^2$?

The zero $x=2$ has multiplicity $3$, and $x=-1$ has multiplicity $2$. Odd multiplicity generally crosses the axis; even multiplicity generally touches and turns.