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If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?
The binomial is a factor of the polynomial.
If a polynomial of degree $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is divided by a binomial of degree 1, what is the degree of the quotient?
For the following exercises, use long division to divide. Specify the quotient and the remainder.
$\left({x}^{2}+5x-1\right)\xf7\left(x-1\right)$
$x+6+\frac{5}{x-1}\text{,}\text{\hspace{0.17em}}\text{quotient:}\text{\hspace{0.17em}}x+6\text{,}\text{\hspace{0.17em}}\text{remainder:}\text{\hspace{0.17em}}\text{5}$
$\left(2{x}^{2}-9x-5\right)\xf7\left(x-5\right)$
$\left(3{x}^{2}+23x+14\right)\xf7\left(x+7\right)$
$3x+2\text{,}\text{\hspace{0.17em}}\text{quotient:}3x+2\text{,}\text{\hspace{0.17em}}\text{remainder:0}$
$\left(4{x}^{2}-10x+6\right)\xf7\left(4x+2\right)$
$\left(6{x}^{2}-25x-25\right)\xf7\left(6x+5\right)$
$x-5\text{,}\text{\hspace{0.17em}}\text{quotient:}\text{\hspace{0.17em}}x-5\text{,}\text{\hspace{0.17em}}\text{remainder:}\text{\hspace{0.17em}}\text{0}$
$\left(-{x}^{2}-1\right)\xf7\left(x+1\right)$
$\left(2{x}^{2}-3x+2\right)\xf7\left(x+2\right)$
$2x-7+\frac{16}{x+2}\text{,}\text{\hspace{0.17em}}\text{quotient:}\text{}\text{\hspace{0.17em}}2x-7\text{,}\text{\hspace{0.17em}}\text{remainder:}\text{\hspace{0.17em}}\text{16}$
$\left({x}^{3}-126\right)\xf7\left(x-5\right)$
$\left(3{x}^{2}-5x+4\right)\xf7\left(3x+1\right)$
$x-2+\frac{6}{3x+1}\text{,}\text{\hspace{0.17em}}\text{quotient:}\text{\hspace{0.17em}}x-2\text{,}\text{\hspace{0.17em}}\text{remainder:}\text{\hspace{0.17em}}\text{6}$
$\left({x}^{3}-3{x}^{2}+5x-6\right)\xf7\left(x-2\right)$
$\left(2{x}^{3}+3{x}^{2}-4x+15\right)\xf7\left(x+3\right)$
$2{x}^{2}-3x+5\text{,}\text{\hspace{0.17em}}\text{quotient:}\text{\hspace{0.17em}}2{x}^{2}-3x+5\text{,}\text{\hspace{0.17em}}\text{remainder:}\text{\hspace{0.17em}}\text{0}$
For the following exercises, use synthetic division to find the quotient.
$\left(3{x}^{3}-2{x}^{2}+x-4\right)\xf7\left(x+3\right)$
$\left(2{x}^{3}-6{x}^{2}-7x+6\right)\xf7(x-4)$
$2{x}^{2}+2x+1+\frac{10}{x-4}$
$\left(6{x}^{3}-10{x}^{2}-7x-15\right)\xf7(x+1)$
$\left(4{x}^{3}-12{x}^{2}-5x-1\right)\xf7(2x+1)$
$2{x}^{2}-7x+1-\frac{2}{2x+1}$
$\left(9{x}^{3}-9{x}^{2}+18x+5\right)\xf7(3x-1)$
$\left(3{x}^{3}-2{x}^{2}+x-4\right)\xf7\left(x+3\right)$
$3{x}^{2}-11x+34-\frac{106}{x+3}$
$\left(-6{x}^{3}+{x}^{2}-4\right)\xf7\left(2x-3\right)$
$\left(2{x}^{3}+7{x}^{2}-13x-3\right)\xf7\left(2x-3\right)$
${x}^{2}+5x+1$
$\left(3{x}^{3}-5{x}^{2}+2x+3\right)\xf7(x+2)$
$\left(4{x}^{3}-5{x}^{2}+13\right)\xf7(x+4)$
$4{x}^{2}-21x+84-\frac{323}{x+4}$
$\left({x}^{3}-3x+2\right)\xf7\left(x+2\right)$
$\left({x}^{3}-21{x}^{2}+147x-343\right)\xf7\left(x-7\right)$
${x}^{2}-14x+49$
$\left({x}^{3}-15{x}^{2}+75x-125\right)\xf7\left(x-5\right)$
$\left(9{x}^{3}-x+2\right)\xf7\left(3x-1\right)$
$3{x}^{2}+x+\frac{2}{3x-1}$
$\left(6{x}^{3}-{x}^{2}+5x+2\right)\xf7\left(3x+1\right)$
$\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\right)\xf7\left(x+1\right)$
${x}^{3}-3x+1$
$\left({x}^{4}-3{x}^{2}+1\right)\xf7\left(x-1\right)$
$\left({x}^{4}+2{x}^{3}-3{x}^{2}+2x+6\right)\xf7\left(x+3\right)$
${x}^{3}-{x}^{2}+2$
$\left({x}^{4}-10{x}^{3}+37{x}^{2}-60x+36\right)\xf7\left(x-2\right)$
$\left({x}^{4}-8{x}^{3}+24{x}^{2}-32x+16\right)\xf7\left(x-2\right)$
${x}^{3}-6{x}^{2}+12x-8$
$\left({x}^{4}+5{x}^{3}-3{x}^{2}-13x+10\right)\xf7\left(x+5\right)$
$\left({x}^{4}-12{x}^{3}+54{x}^{2}-108x+81\right)\xf7\left(x-3\right)$
${x}^{3}-9{x}^{2}+27x-27$
$\left(4{x}^{4}-2{x}^{3}-4x+2\right)\xf7\left(2x-1\right)$
$\left(4{x}^{4}+2{x}^{3}-4{x}^{2}+2x+2\right)\xf7\left(2x+1\right)$
$2{x}^{3}-2x+2$
For the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it is, indicate the factorization.
$x-2,\text{\hspace{0.17em}}4{x}^{3}-3{x}^{2}-8x+4$
$x-2,\text{\hspace{0.17em}}3{x}^{4}-6{x}^{3}-5x+10$
Yes $\text{\hspace{0.17em}}\left(x-2\right)(3{x}^{3}-5)$
$x+3,\text{\hspace{0.17em}}-4{x}^{3}+5{x}^{2}+8$
$x-2,\text{\hspace{0.17em}}4{x}^{4}-15{x}^{2}-4$
Yes $\text{\hspace{0.17em}}\left(x-2\right)(4{x}^{3}+8{x}^{2}+x+2)$
$x-\frac{1}{2},\text{\hspace{0.17em}}2{x}^{4}-{x}^{3}+2x-1$
$x+\frac{1}{3},\text{\hspace{0.17em}}3{x}^{4}+{x}^{3}-3x+1$
No
For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.
Factor is $\text{\hspace{0.17em}}{x}^{2}-x+3$
Factor is $\text{\hspace{0.17em}}({x}^{2}+2x+4)$
$(x-1)({x}^{2}+2x+4)$
Factor is $\text{\hspace{0.17em}}{x}^{2}+2x+5$
Factor is $\text{\hspace{0.17em}}{x}^{2}+x+1$
$(x-5)({x}^{2}+x+1)$
Factor is ${x}^{2}+2x+2$
For the following exercises, use synthetic division to find the quotient and remainder.
$\frac{4{x}^{3}-33}{x-2}$
$\text{Quotient:}\text{\hspace{0.17em}}4{x}^{2}+8x+16\text{,}\text{\hspace{0.17em}}\text{remainder:}\text{\hspace{0.17em}}-1$
$\frac{2{x}^{3}+25}{x+3}$
$\frac{3{x}^{3}+2x-5}{x-1}$
$\text{Quotient:}\text{\hspace{0.17em}}3{x}^{2}+3x+5\text{,}\text{\hspace{0.17em}}\text{remainder:}\text{\hspace{0.17em}}0$
$\frac{-4{x}^{3}-{x}^{2}-12}{x+4}$
$\frac{{x}^{4}-22}{x+2}$
$\text{Quotient:}\text{\hspace{0.17em}}{x}^{3}-2{x}^{2}+4x-8\text{,}\text{\hspace{0.17em}}\text{remainder:}\text{\hspace{0.17em}}-6$
For the following exercises, use a calculator with CAS to answer the questions.
Consider $\text{\hspace{0.17em}}\frac{{x}^{k}-1}{x-1}\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}k=1,2,3.\text{\hspace{0.17em}}$ What do you expect the result to be if $\text{\hspace{0.17em}}k=4?$
Consider $\text{\hspace{0.17em}}\frac{{x}^{k}+1}{x+1}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}k=1,3,5.\text{\hspace{0.17em}}$ What do you expect the result to be if $\text{\hspace{0.17em}}k=7?$
${x}^{6}-{x}^{5}+{x}^{4}-{x}^{3}+{x}^{2}-x+1$
Consider $\text{\hspace{0.17em}}\frac{{x}^{4}-{k}^{4}}{x-k}\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}k=1,2,3.\text{\hspace{0.17em}}$ What do you expect the result to be if $\text{\hspace{0.17em}}k=4?$
Consider $\text{\hspace{0.17em}}\frac{{x}^{k}}{x+1}\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}k=1,2,3.\text{\hspace{0.17em}}$ What do you expect the result to be if $\text{\hspace{0.17em}}k=4?$
${x}^{3}-{x}^{2}+x-1+\frac{1}{x+1}$
Consider $\text{\hspace{0.17em}}\frac{{x}^{k}}{x-1}\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}k=1,2,3.\text{\hspace{0.17em}}$ What do you expect the result to be if $\text{\hspace{0.17em}}k=4?$
For the following exercises, use synthetic division to determine the quotient involving a complex number.
$\frac{{x}^{2}+1}{x-i}$
$\frac{{x}^{2}+1}{x+i}$
For the following exercises, use the given length and area of a rectangle to express the width algebraically.
Length is $\text{\hspace{0.17em}}x+5,\text{\hspace{0.17em}}$ area is $\text{\hspace{0.17em}}2{x}^{2}+9x-5.$
Length is $\text{\hspace{0.17em}}2x\text{}+\text{}5,\text{\hspace{0.17em}}$ area is $\text{\hspace{0.17em}}4{x}^{3}+10{x}^{2}+6x+15$
$2{x}^{2}+3$
Length is $\text{\hspace{0.17em}}3x\u20134,\text{\hspace{0.17em}}$ area is $\text{\hspace{0.17em}}6{x}^{4}-8{x}^{3}+9{x}^{2}-9x-4$
For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.
Volume is $\text{\hspace{0.17em}}12{x}^{3}+20{x}^{2}-21x-36,\text{\hspace{0.17em}}$ length is $\text{\hspace{0.17em}}2x+3,\text{\hspace{0.17em}}$ width is $\text{\hspace{0.17em}}3x-4.$
$2x+3$
Volume is $\text{\hspace{0.17em}}18{x}^{3}-21{x}^{2}-40x+48,\text{\hspace{0.17em}}$ length is $\text{\hspace{0.17em}}3x\u20134,\text{\hspace{0.17em}}$ width is $\text{\hspace{0.17em}}3x\u20134.$
Volume is $\text{\hspace{0.17em}}10{x}^{3}+27{x}^{2}+2x-24,\text{\hspace{0.17em}}$ length is $\text{\hspace{0.17em}}5x\u20134,\text{\hspace{0.17em}}$ width is $\text{\hspace{0.17em}}2x+3.$
$x+2$
Volume is $\text{\hspace{0.17em}}10{x}^{3}+30{x}^{2}-8x-24,\text{\hspace{0.17em}}$ length is $\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}$ width is $\text{\hspace{0.17em}}x+3.$
For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.
Volume is $\text{\hspace{0.17em}}\pi (25{x}^{3}-65{x}^{2}-29x-3),\text{\hspace{0.17em}}$ radius is $\text{\hspace{0.17em}}5x+1.$
$x-3$
Volume is $\text{\hspace{0.17em}}\pi (4{x}^{3}+12{x}^{2}-15x-50),\text{\hspace{0.17em}}$ radius is $\text{\hspace{0.17em}}2x+5.$
Volume is $\text{\hspace{0.17em}}\pi (3{x}^{4}+24{x}^{3}+46{x}^{2}-16x-32),\text{\hspace{0.17em}}$ radius is $\text{\hspace{0.17em}}x+4.$
$3{x}^{2}-2$
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