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Chapter 5: Problem 48

Factor each trinomial completely. $$ 9 t^{2}+24 t r+16 r^{2} $$

Short Answer

Expert verified
\(9t^2 + 24tr + 16r^2\) factors to \((3t + 4r)^2\).

Step by step solution

01

- Identify coefficients

Identify the coefficients of the trinomial. The given trinomial is in the form of \(9t^2 + 24tr + 16r^2\). Here, the coefficient of \(t^2\) is 9, the coefficient of \(tr\) is 24, and the coefficient of \(r^2\) is 16.
02

- Recognize a perfect square trinomial

Observe that the trinomial is a perfect square trinomial because it can be written in the form of \((at + br)^2\). Here, \(a^2 = 9\), \(b^2 = 16\), and \(ab = 12\), so \(2ab = 24\).
03

- Write the trinomial as a squared binomial

Since the trinomial matches the form of a perfect square trinomial, write it as the square of a binomial. The trinomial \(9t^2 + 24tr + 16r^2\) factors to \((3t + 4r)^2\).
04

- Final factorized form

The final factorized form of the trinomial is \((3t + 4r)(3t + 4r)\) which is written as \((3t + 4r)^2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

perfect square trinomial
A perfect square trinomial is one that can be expressed as the square of a binomial. This means the trinomial looks like \[a^2 + 2ab + b^2 = (a+b)^2\]. In our example, the trinomial \[9t^2 + 24tr + 16r^2\] fits this pattern. Notice the following: \[9t^2 = (3t)^2\], \[16r^2 = (4r)^2\], and \[24tr = 2 \cdot 3t \cdot 4r\]. This confirms that \[9t^2 + 24tr + 16r^2\] is indeed a perfect square trinomial.
factoring polynomials
Factoring polynomials helps us break them down into simpler components. This makes solving equations easier. In this case, we recognized our polynomial as a perfect square trinomial. The steps to factor it involved identifying the coefficients and realizing they fit the form of \[a^2 + 2ab + b^2\]. By understanding this pattern, we immediately knew \[9t^2 + 24tr + 16r^2\] could be written as \[(3t + 4r)^2\]. Factoring isn't always this straightforward, but recognizing patterns can help you a lot.
squared binomial
A squared binomial is an expression of the form \[(a+b)^2\]. When you expand \[(3t + 4r)^2\], you get: \[ (3t + 4r)(3t + 4r) = 3t(3t + 4r) + 4r(3t + 4r) = 9t^2 + 12tr + 12tr + 16r^2 = 9t^2 + 24tr + 16r^2\]. This shows that squaring the binomial \[3t + 4r\] yields our original trinomial. The ability to recognize and write a trinomial as a squared binomial is a key skill in algebra.

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Most popular questions from this chapter

Factor each trinomial completely. $$ 2 x^{2}+24 x+72 $$

Factor polynomial. \(z^{10}-4 z^{9} y-21 z^{8} y^{2}\)

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Students often have difficulty when factoring by grouping because they are not able to tell when the polynomial is completely factored. For example, $$5 y(2 x-3)+8 t(2 x-3)$$ is not in factored form, because it is the sum of two terms: \(5 y(2 x-3)\) and \(8 t(2 x-3)\) However, because \(2 x-3\) is a common factor of these two terms, the expression can now be factored. $$(2 x-3)(5 y+8 t)$$ The factored form is a product of two factors: \(2 x-3\) and \(5 y+8 t\) Determine whether each expression is in factored form or is not in factored form. If it is not in factored form, factor it if possible. $$ 18 x^{2}(y+4)+7(y-4) $$
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