Polynomials and Factoring

1)       $3x^2(5x-1) + 2x(x+3) =$

$15x^3 + 5x^2 + 6x$

$17x^2 + 6x - 1$

$15x^3 - x^2 + 6x$

$15x^3 + 2x^2 + 6x - 1$

$12x^2 + 8x$

Hint Use the distributive law $a(b+c)=ab+ac$ and combine like terms.

2)       $x=3$ is a root of

$y = x^3 + 3x^2 - x - 3$

$y = x^3 - x^2 - 14x + 24$

$y = x^3 - 6x^2 - x + 6$

$y = x^3 + 3x^2 - 4x - 12$

$y = x^3 + 6x^2 + 11x + 6$

Hint $x=a$ is a root of $p(x)$ if $p(a)=0$.

3)       The zeros of $y = 2x^2 - 5x - 12$ are $x = $

$\frac{-3}{2}$ and $4$

$\frac{-2}{3}$ and $4$

$-2$ and $3$

$-6$ and $\frac{1}{2}$

undefined (there are no roots)

Hint Factor into linear terms of the form $(px-r)(qx-s)$. Then the roots are $x=r/p, x=s/q$.

4)       All polynomials of degree three have

three linear factors and one $y$-intercept.*

three or two or one linear factor(s) and one $y$-intercept.*

three or one linear factor and no $y$-intercept.*

three or one linear factor and one $y$-intercept.*

three or one or no linear factors where some will have a $y$-intercept and some will not.*

* Factors need not be distinct.
Hint 1 A polynomial of degree three has at most three roots and intersects the $x$-axis at least once.
Hint 2 The $y$-intercept of a polynomial $p(x)$ is at $p(0)$.

5)       $(u^2-7u+2)(3v^2-5)=$

$3u^2v^2 - 6u^2 - 21uv^2 + 30u + 6v^2 - 7$

$3u^2v^2 + 5u^2 - 21uv^2 + 35u + 6v^2 - 10$

$3u^2v^2 - 5u^2 - 21uv^2 + 35u + 6v^2 - 10$

$3u^2v^2 - 5u^2 - 22uv^2 + 35u + 6v^2 - 10$

$3u^2v^2 - 5u^2 + 21uv^2 - 35u + 6v^2 - 10$

Hint Multiply each term in the first parentheses with each term in the second parentheses and add up all the resulting terms.

6)       The two polynomials $x^3-6x^2+11x-6$ and $x^3-8x^2+20x-16$ have:

$(x-1)$ as a common factor.

$(x-2)$ as a common factor.

$(x-3)$ as a common factor.

$(x-4)$ as a common factor.

no common factors

Hint A polynomial $p(x)$ has $(x-a)$ as a factor if and only if $p(a)=0$.

7)       $\frac{x^4+x^3+x^2-x-1}{x^2+2}=$

$-x+1$ with remainder $x^2-1$

$-3x+1$ with remainder $x^2+x-1$

$x^4+x^3-x-3$ with remainder $0$

$x^2-1$ with remainder $-x+1$

$x^2+x-1$ with remainder $-3x+1$

Hint Use long division.

8)       If $p(2) = 0$, then

$p(x)$ has $(x + 2)$ as a factor.

$p(x)$ has $(x - 2)$ as a factor.

the constant term of $p(x)$ is $-2$.

$p(x)$ must have odd degree.

$p(x)$ does not necessarily have any $x$-intercepts.

Hint A polynomial $p(x)$ has $(x-a)$ as a factor if and only if $p(a)=0$.

9)       After factoring $12x^2+5x-3$, we get:

$(4x-3)(3x+1)$

$(4x+3)(3x-1)$

$(2x+3)(6x-1)$

$(6x+3)(2x-1)$

$(2x+1)(6x-3)$

Hint Notice that after factoring $ax^2+bx+c=(px+r)(qx+s)=pqx^2+(qr+ps)x+rs$ so we'll have $pq=a$, $rs=c$.

10)       $\frac{y^3-27}{y-3}=$

$y^2 + 3y + 9$

$y^2 + 9$

$y^2 + 3$

$y^2 - 3y + 9$

$y^2 + 3y - 9$

Hint Use the formula $A^3-B^3=(A-B)(A^2+AB+B^2)$ and cancel common factors.