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Factor a. $\text{\hspace{0.17em}}2{x}^{2}+9x+9\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}6{x}^{2}+x-1$
a. $\text{\hspace{0.17em}}(2x+3)(x+3)\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}\left(3x\mathrm{-1}\right)\left(2x+1\right)$
A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.
We can use this equation to factor any perfect square trinomial.
A perfect square trinomial can be written as the square of a binomial:
Given a perfect square trinomial, factor it into the square of a binomial.
Factor $\text{\hspace{0.17em}}25{x}^{2}+20x+4.$
Notice that $\text{\hspace{0.17em}}25{x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ are perfect squares because $\text{\hspace{0.17em}}25{x}^{2}={(5x)}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}4={2}^{2}.\text{\hspace{0.17em}}$ Then check to see if the middle term is twice the product of $\text{\hspace{0.17em}}5x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ The middle term is, indeed, twice the product: $\text{\hspace{0.17em}}2(5x)(2)=20x.\text{\hspace{0.17em}}$ Therefore, the trinomial is a perfect square trinomial and can be written as $\text{\hspace{0.17em}}{(5x+2)}^{2}.$
Factor $\text{\hspace{0.17em}}49{x}^{2}-14x+1.$
${(7x\mathrm{-1})}^{2}$
A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.
We can use this equation to factor any differences of squares.
A difference of squares can be rewritten as two factors containing the same terms but opposite signs.
Given a difference of squares, factor it into binomials.
Factor $\text{\hspace{0.17em}}9{x}^{2}-25.$
Notice that $\text{\hspace{0.17em}}9{x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}25\text{\hspace{0.17em}}$ are perfect squares because $\text{\hspace{0.17em}}9{x}^{2}={(3x)}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}25={5}^{2}.\text{\hspace{0.17em}}$ The polynomial represents a difference of squares and can be rewritten as $\text{\hspace{0.17em}}(3x+5)(3x-5).$
Factor $\text{\hspace{0.17em}}81{y}^{2}-100.$
$(9y+10)(9y-10)$
Is there a formula to factor the sum of squares?
No. A sum of squares cannot be factored.
Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.
Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.
We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: S ame O pposite A lways P ositive. For example, consider the following example.
The sign of the first 2 is the same as the sign between $\text{\hspace{0.17em}}{x}^{3}-{2}^{3}.\text{\hspace{0.17em}}$ The sign of the $\text{\hspace{0.17em}}2x\text{\hspace{0.17em}}$ term is opposite the sign between $\text{\hspace{0.17em}}{x}^{3}-{2}^{3}.\text{\hspace{0.17em}}$ And the sign of the last term, 4, is always positive .
We can factor the sum of two cubes as
We can factor the difference of two cubes as
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