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For the following polynomials, identify the degree, the leading term, and the leading coefficient.
Identify the degree, leading term, and leading coefficient of the polynomial $\text{\hspace{0.17em}}4{x}^{2}-{x}^{6}+2x-6.$
The degree is 6, the leading term is $\text{\hspace{0.17em}}-{x}^{6},$ and the leading coefficient is $\text{\hspace{0.17em}}\mathrm{-1.}$
We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, $\text{\hspace{0.17em}}5{x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{-2}{x}^{2}\text{\hspace{0.17em}}$ are like terms, and can be added to get $\text{\hspace{0.17em}}3{x}^{2},$ but $\text{\hspace{0.17em}}3x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}3{x}^{2}\text{\hspace{0.17em}}$ are not like terms, and therefore cannot be added.
Given multiple polynomials, add or subtract them to simplify the expressions.
Find the sum.
$\left(12{x}^{2}+9x-21\right)+\left(4{x}^{3}+8{x}^{2}-5x+20\right)$
$$\begin{array}{cc}4{x}^{3}+\left(12{x}^{2}+8{x}^{2}\right)+(9x-5x)+(\mathrm{-21}+20)\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Combineliketerms}.\hfill \\ 4{x}^{3}+20{x}^{2}+4x-1\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Simplify}.\hfill \end{array}$$
Find the sum.
$\left(2{x}^{3}+5{x}^{2}-x+1\right)+\left(2{x}^{2}-3x-4\right)$
$2{x}^{3}+7{x}^{2}\mathrm{-4}x\mathrm{-3}$
Find the difference.
$\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)$
$$\begin{array}{cc}7{x}^{4}-5{x}^{3}+\left(-{x}^{2}+2{x}^{2}\right)+(6x-3x)+(1-2)\text{}\hfill & \phantom{\rule{1em}{0ex}}\text{Combineliketerms}.\hfill \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x-1\hfill & \phantom{\rule{1em}{0ex}}\text{Simplify}.\hfill \end{array}$$
Find the difference.
$\left(\mathrm{-7}{x}^{3}-7{x}^{2}+6x-2\right)-\left(4{x}^{3}-6{x}^{2}-x+7\right)$
$\mathrm{-11}{x}^{3}-{x}^{2}+7x\mathrm{-9}$
Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.
To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ in $\text{\hspace{0.17em}}2(x+7)\text{\hspace{0.17em}}$ to obtain the equivalent expression $\text{\hspace{0.17em}}2x+14.\text{\hspace{0.17em}}$ When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.
Given the multiplication of two polynomials, use the distributive property to simplify the expression.
Find the product.
$\left(2x+1\right)\left(3{x}^{2}-x+4\right)$
$$\begin{array}{cc}2x\left(3{x}^{2}-x+4\right)+1\left(3{x}^{2}-x+4\right)\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Usethedistributiveproperty}.\hfill \\ \left(6{x}^{3}-2{x}^{2}+8x\right)+\left(3{x}^{2}-x+4\right)\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Multiply}.\hfill \\ 6{x}^{3}+\left(\mathrm{-2}{x}^{2}+3{x}^{2}\right)+(8x-x)+4\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Combineliketerms}.\hfill \\ 6{x}^{3}+{x}^{2}+7x+4\hfill & \phantom{\rule{2em}{0ex}}\text{\hspace{1em}\hspace{1em}}\text{Simplify}.\hfill \end{array}$$
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