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Chapter 6: Problem 1

For exercises 1-20, use a pattern to factor. Check. Identify any prime polynomials. $$ x^{2}+8 x+16 $$

Short Answer

Expert verified
(x + 4)^2, not prime.

Step by step solution

01

- Identify the Standard Form

The given polynomial is in the standard quadratic form: \(ax^2 + bx + c\), where \(a = 1\), \(b = 8\), and \(c = 16\).
02

- Apply the Factoring Pattern for Perfect Square Trinomials

A perfect square trinomial can be factored using the pattern: \(a^2 + 2ab + b^2 = (a + b)^2\). Here, we check if the given polynomial fits this pattern. Since \(x^2 + 8x + 16\) can be written as \((x)^2 + 2(x)(4) + (4)^2\), it matches the perfect square trinomial pattern with \(a = x\) and \(b = 4\).
03

- Write the Factored Form

Using the perfect square trinomial pattern, the polynomial can be factored as: \((x + 4)^2\).
04

- Verify the Factored Form

Re-expanding \((x + 4)^2\) should yield the original polynomial: \((x + 4)(x + 4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16\). Thus, the factorization is correct.
05

- Determine if the Polynomial is Prime

A polynomial is prime if it cannot be factored further using integers. Since \((x + 4)^2\) is a valid factorization, the polynomial \(x^2 + 8x + 16\) is not prime.

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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 specific type of quadratic polynomial. It has the form differentiated from common quadratics. It takes the form: a^2 + 2ab + b^2. This pattern always factors into (a + b)^2. To identify a perfect square trinomial:
  • Check if both the first and last terms can be represented as squares.
  • Check if the middle term is twice the product of the terms.
For instance, in the polynomial: x^2 + 8x + 16, we see:x^2 is a square (x squared), 4^2 is a square (16), and 8x is 2 x 4. Therefore, it factors to (x + 4)^2.
Quadratic Equations
Quadratic equations are polynomials of degree 2. They follow the general form: ax^2 + bx + c = 0. Solutions for quadratics include factoring, the quadratic formula, or completing the square. For easy understanding,
  • 'a' is the coefficient of x^2.
  • 'b' is the coefficient of x.
  • 'c' is the constant term.
For example, if you have x^2 + 8x + 16 ,it is already in standard form.In this case: a = 1, b = 8 and c = 16.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into products of simpler polynomials. It's like finding components that multiply back to the original form. For quadratic polynomials like x^2 + 8x + 16, factoring involves identifying patterns or using formulas. Patterns include:
  • Difference of squares.
  • Perfect square trinomials.
  • Common monomial factors.
Always verify your factors by expanding them. For (x + 4)^2, expand back to x^2 + 8x + 16.
Standard Quadratic Form
Polynomials should be in standard quadratic form for ease of solving. This form is: ax^2 + bx + c. Let's take a closer look at a polynomial: x^2 + 8x + 16. This structure helps tremendously when using different solving techniques. Note the coefficients a, b, and c:

  • a (coefficient of x^2) helps in judging if factorizable.
  • b (coefficient of x) determines the middle term.
  • c (constant term) provides the number to look at for product pairs.
For the exercise, we have x^2 + 8x + 16easily fits into ax^2 + bx + c. With a clear structure, it's easy to identify it as a perfect square trinomial.

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Most popular questions from this chapter

A student said that the solutions of \((x+2)(x-6)=15\) are \(x=-2\) and \(x=6\), since \(-2+2=0\) and \(6-6=0\). Explain what is wrong with this thinking.

(a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Factor \(45 c^{3}+33 c^{2}-168 c\) completely. Incorrect Answer: $$ \begin{aligned} &45 c^{3}+33 c^{2}-168 c \\ &=3 c\left(15 c^{2}+11 c-56\right) \end{aligned} $$

Factor completely. Identify any prime polynomials. $$ 11 h^{3}-88 k^{3} $$

Solve. $$ (c+3) c=0 $$

Factor completely. Identify any prime polynomials. $$ 16 p^{3}-8 p^{2}+p $$
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