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Chapter 5: Problem 165

Factor each polynomial using the trial-and-error method. $$ 5 y^{2}+16 y+3 $$

Short Answer

Expert verified
The factorization is \[(5y + 1)(y + 3)\].

Step by step solution

01

Write the polynomial in standard form

The polynomial is already in standard form: \[5y^2 + 16y + 3\]
02

Identify coefficients

Identify the coefficients of the polynomial: \[a = 5, b = 16, c = 3\]
03

Set up trial factors

Set up potential factor pairs of the quadratic term and the constant term. For example: - Factors of 5 are (5, 1) and (-5, -1). - Factors of 3 are (3, 1), (-3, -1), (1, 3), and (-1, -3).
04

Test factor pairs

Test the different combinations to find the one that satisfies both the middle term and the constant term: \[(5y + 1)(y + 3)\]Check: \[(5y + 1)(y + 3) = 5y^2 + 15y + y + 3 = 5y^2 + 16y + 3\]
05

Verify the factorization

Verify that multiplying the factors gives the original polynomial: \[5y^2 + 16y + 3\]. The factorization is correct.

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Key Concepts

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

Quadratic Equations
Quadratic equations are a type of polynomial equation of degree 2. They typically take the form:
  • \( ax^2 + bx + c = 0 \)
where \(a\), \(b\), and \(c\) are coefficients and \(a eq 0\).The solutions to quadratic equations are the values of \(x\) that make the equation true.There are several methods to solve quadratic equations, one of which is factoring.In this article, we will explore factoring using a specific technique called the trial-and-error method.
Trial-and-Error Method in Factoring
The trial-and-error method is a strategy used to factor quadratic polynomials.It involves testing different combinations of factors to identify the correct ones that will simplify the polynomial.Here are the basic steps in using the trial-and-error method to factor a quadratic polynomial:
  • Identify the coefficients in the polynomial. For example, if you have \(5y^2 + 16y + 3\), the coefficients are: \(a = 5\), \(b = 16\), and \(c = 3\).
  • List out possible factor pairs of the constant term \(c\) and the quadratic term \(a\). For \(c\), possible pairs are: (3, 1), (-3, -1), (1, 3), and (-1, -3). For \(a\), possible pairs are: (5, 1) and (-5, -1).
  • Test the different combinations by multiplying them and comparing the result to the original quadratic polynomial. For example:
    • Test: \[(5y + 1)(y + 3)\]
    • Calculation: \[(5y)(y) + (5y)(3) + (1)(y) + (1)(3)\]
    • Result: \[5y^2 + 16y + 3\]
    • This matches the original polynomial, so the correct factorization is \[(5y + 1)(y + 3)\].
Using trial-and-error requires patience and practice, but it can be a highly effective method for factoring quadratic polynomials.
Polynomial Coefficients
Polynomial coefficients are the numerical values placed in front of the variables in polynomial terms.
  • In a general polynomial, such as \(5y^2 +16y + 3\), the coefficients are 5, 16, and 3.
  • The coefficient of the quadratic term \(y^2\) is 5.
  • The coefficient of the linear term \(y\) is 16.
  • The constant term, 3, is also considered a coefficient.
Coefficients play a crucial role in factoring and solving polynomials as they determine the values that need to be paired together to simplify the expression.Understanding and identifying coefficients is the first step in successful polynomial factoring and manipulation.

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Most popular questions from this chapter

Solve each equation. $$(x-1)(x+3)(x-9)=0$$

Factor each polynomial by grouping. $$ x^{2} a^{2}+a^{2}+x^{2}+1 $$

Factor each polynomial using the trial-and-error method. $$ a^{2}-9 a+18 $$

Factor each polynomial completely. $$ x^{2} z+2 x y z+y^{2} z $$

Tossing a ball. A boy tosses a ball upward at 32 feet per second from a window that is 48 feet above the ground. The height of the ball above the ground (in feet) at time \(t\) (in seconds) is given by $$h(t)=-16 t^{2}+32 t+48$$ Find the time at which the ball strikes the ground.
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