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Chapter 6: Problem 74

Factor completely. If a polynomial is prime, state this. $$ 9 c^{2}+6 c d+d^{2} $$

Short Answer

Expert verified
The polynomial factors to \((3c + d)^2\).

Step by step solution

01

Recognize the Form

Identify if the polynomial fits a recognizable pattern. Notice that the polynomial \(9c^2 + 6cd + d^2\) fits the perfect square trinomial form \(a^2 + 2ab + b^2\). This particular form can be factored into \((a + b)^2\).
02

Identify Parts of the Trinomial

Compare \(9c^2 + 6cd + d^2\) with \(a^2 + 2ab + b^2\). We can see that \(a^2 = 9c^2\), \(2ab = 6cd\), and \(b^2 = d^2\). Therefore, \(a = 3c\) and \(b = d\).
03

Write the Factorization

Using the values of \(a\) and \(b\) found in the previous step, rewrite the polynomial as \((3c + d)^2\).
04

Verify the Factorization

Expand \((3c + d)^2\) to verify that it equals the original polynomial: \((3c + d)^2 = 9c^2 + 6cd + d^2\). This confirms the factorization is correct.

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Key Concepts

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

Perfect square trinomials
A perfect square trinomial is a special type of polynomial. It occurs when a polynomial can be written as a binomial squared. Generally, the form is represented as \( a^2 + 2ab + b^2 \). This can always be factored into \( (a + b)^2 \).
In the given polynomial, \(9c^2 + 6cd + d^2 \), we can see it fits this form:
  • \(a^2 = 9c^2\), which means \(a = 3c\)
  • \(b^2 = d^2\), so \(b = d\)
  • \(2ab = 6cd\), confirming \(2 \times 3c \times d = 6cd\)
By identifying these parts, we recognize it as a perfect square trinomial, allowing us to factor it as \((3c + d)^2\).
Polynomial factorization
Polynomial factorization is the process of breaking down a polynomial into simpler 'factors' that, when multiplied together, give back the original polynomial. Understanding patterns like perfect square trinomials makes this easier.
In the polynomial \(9c^2 + 6cd + d^2\), we saw it matches the perfect square trinomial form. Thus, we factor it step-by-step:
  • Identify potential values for \(a\) and \(b\) : \(a = 3c\) and \(b = d\)
  • Rewrite the polynomial: \((3c + d)^2\)
Verification involves expanding \((3c + d)^2\) to check if it equals the original polynomial:
\((3c + d)(3c + d) = 9c^2 + 6cd + d^2\). Thus, the factorization is correct.
Quadratic expressions
Quadratic expressions are polynomials of degree two. They have the general form \(ax^2 + bx + c \). Sometimes, they can be factored if they fit specific patterns, such as perfect square trinomials.
In our case, \(9c^2 + 6cd + d^2\) is a quadratic expression and fits the perfect square trinomial pattern. By using the aforementioned steps, we successfully factored it into \((3c + d)^2\).
Knowing how to handle different forms of quadratic expressions can greatly simplify your work in algebra and beyond.
Always remember to:
  • Look for special patterns
  • Break down the problem
  • Verify your solutions
By mastering these approaches, tackling quadratic expressions becomes much more manageable.

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Most popular questions from this chapter

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ -80 a^{6}+45 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ m^{2}-64 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ p^{2} q-25 q+3 p^{2}-75 $$

Review multiplying binomials using FOIL $$ (x+2)(x-7) $$

Let \(g(x)=x^{2} .\) Find \(a\) such that \(g(a)=144\).
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