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Q12.

Expert-verifiedFound in: Page 523

Book edition
Student Edition

Author(s)
Carter, Cuevas, Day, Holiday, Luchin

Pages
801 pages

ISBN
9780078884801

**Determine whether each trinomial is a perfect square trinomial****. Write yes or no. If so, factor it.**

**$4{x}^{2}+28x+49$**

**Yes**, the given trinomial is a perfect square trinomial.

The factorization of the given trinomial is ${\left(2x+7\right)}^{\mathbf{2}}$.

The given trinomial is: $4{x}^{2}+28x+49$

The First, middle and last terms of the given trinomial are $4{x}^{2},28x$ and 49 respectively.

The first term of the given trinomial can be written as:

$4{x}^{2}={\left(2x\right)}^{2}$

Therefore, the first term of the given trinomial is a perfect square.

The last term of the given trinomial can be written as:

$49={\left(7\right)}^{2}$

Therefore, the last term of the given trinomial is a perfect square.

The middle term of the given trinomial can be written as:

$28x=2\left(2x\right)\left(7\right)$

Therefore, the middle term is twice the product of the square roots of the first term and last term.

As the first and last terms of the given trinomial are a perfect square and the middle term is twice the product of the square roots of the first term and last term.

Therefore, yes, the given trinomial is a perfect square trinomial.

It is known that:

${a}^{2}+2ab+{b}^{2}=\left(a+b\right)\left(a+b\right)={\left(a+b\right)}^{2}$

It can be noticed that:

$4{x}^{2}+28x+49={\left(2x\right)}^{2}+2\left(2x\right)\left(7\right)+{\left(7\right)}^{2}\phantom{\rule{0ex}{0ex}}=\left(2x+7\right)\left(2x+7\right)\text{}\left(\because \text{}{a}^{2}+2ab+{b}^{2}=\left(a+b\right)\left(a+b\right)={\left(a+b\right)}^{2}\right)\phantom{\rule{0ex}{0ex}}={\left(2x+7\right)}^{2}$

Therefore, the factorization of the given trinomial is ${\left(2x+7\right)}^{2}$.

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