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Chapter 5: Problem 73

Factor each polynomial completely. See Examples 1 through 12. $$ 9 x^{2}+30 x+25 $$

Short Answer

Expert verified
The polynomial \(9x^2 + 30x + 25\) factors to \((3x + 5)^2\).

Step by step solution

01

Identify the Polynomial

We are given the quadratic polynomial \(9x^2 + 30x + 25\). Our goal is to factor this polynomial completely.
02

Check if It Is a Perfect Square Trinomial

We will determine if the polynomial \(9x^2 + 30x + 25\) is a perfect square trinomial. A trinomial \(a^2 + 2ab + b^2\) can be expressed as \((a + b)^2\). Here, we check if the first and last terms are perfect squares and if the middle term is twice the product of the square roots of these terms. The first term \(9x^2\) is \((3x)^2\), and the last term \(25\) is \(5^2\). The middle term \(30x\) needs to be twice the product of \(3x\) and \(5\). As \(2 \times 3x \times 5 = 30x\), the polynomial is a perfect square trinomial.
03

Write the Polynomial as a Square of a Binomial

Since the polynomial is a perfect square trinomial, we can express it as \((3x + 5)^2\). Therefore, the completely factored form of the polynomial \(9x^2 + 30x + 25\) is \((3x + 5)^2\).

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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 a special form of a quadratic polynomial. Trinomials are expressions with three terms, and perfect square trinomials take the form \(a^2 + 2ab + b^2\). Detecting this pattern can simplify factoring significantly.

To check if a trinomial is a perfect square, follow these steps:
  • Ensure the first term is a perfect square. In our example \(9x^2\), which equals \((3x)^2\).
  • Confirm the last term is a perfect square. Here, \(25\) is \(5^2\).
  • The middle term should be twice the product of the square roots of the first and last terms. Thus, \(2 \times 3x \times 5\) should equal \(30x\).
The pattern matches perfectly, so you can confidently state that \(9x^2 + 30x + 25\) is a perfect square trinomial.

Identifying this pattern streamlines the process and allows the trinomial to be written as \((3x + 5)^2\).
Quadratic Polynomials
Quadratic polynomials are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is a variable. The highest power of \(x\) in these expressions is 2.

Understanding their structure helps in solving and factoring them efficiently. Their graph represents a parabola which may open upwards or downwards.
  • The leading coefficient \(a\) affects the width and direction of the parabola.
  • The constant \(c\) impacts the y-intercept of the graph.
  • The linear term \(b\) influences the symmetry axis and vertex position.
Recognizing and working with quadratic polynomials is integral to solving various algebraic equations, both in academic settings and practical scenarios.
Factoring Techniques
Factoring techniques are crucial for simplifying polynomial expressions by breaking them into simpler parts. These techniques allow mathematicians to solve equations, simplify expressions, or find function roots.

Some common factoring techniques include:
  • Factoring out the Greatest Common Factor (GCF): checking if all terms have a common factor and dividing by it.
  • Grouping: applicable when terms can be grouped into pairs that can be factored separately.
  • Quadratic Trinomials: recognizing special patterns such as perfect squares or implementing methods like the quadratic formula.
  • Difference of Squares: utilized when a polynomial fits the pattern \(a^2 - b^2\) and can be factored into \((a-b)(a+b)\).
Knowing these techniques expands your algebraic toolkit, making it easier to handle complex polynomial equations and deduce their factored forms.

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Most popular questions from this chapter

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7\) find the following. \(-5[P(x)]-Q(x)\)

Factor each polynomial completely. See Examples 1 through 12. $$ x^{2}+4 x+5 $$

Find the value of \(c\) that makes each trinomial a perfect square trinomial. $$ x^{2}+c x+36 $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{2 n}+10 x^{n}+16 $$

Factor. Assume that variables used as exponents represent positive integers. $$ x^{4 n}-16 $$
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