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Magazine Chapter 5: Problem 52
Factor. $$ 35 y^{2}+34 y+8 $$
Short Answer
Expert verified
The factorization of 35y虏 + 34y + 8 is (5y + 2)(7y + 4).
Step by step solution
01
- Identify coefficients
The given quadratic expression is: 35y^{2} + 34y + 8 Identify the coefficients a, b, and c in the standard quadratic form ax^2 + bx + c. Here, a = 35, b = 34, and c = 8.
02
- Find the product ac
Multiply the coefficients a and c. ac = 35 * 8 = 280
03
- Find two numbers that multiply to ac and add to b
We need to find two numbers that multiply to 280 (the product ac) and add to 34 (the coefficient b). These numbers are 14 and 20. 14 * 20 = 280 14 + 20 = 34
04
- Rewrite the middle term using the two numbers
Rewrite the middle term (34y) using the two numbers found: 14 and 20. 35y^{2} + 14y + 20y + 8
05
- Factor by grouping
Group the terms in pairs and factor out the greatest common factor (GCF) from each pair: 35y^{2} + 14y + 20y + 8 = 7y(5y + 2) + 4(5y + 2)
06
- Factor out the common binomial factor
Factor out the common binomial factor (5y + 2): 7y(5y + 2) + 4(5y + 2) = (5y + 2)(7y + 4)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Expressions
A quadratic expression is any expression that can be written in the form: ewline ax^2 + bx + c ewline where:
- a, b, and c are constants
- x is a variable
Factoring by Grouping
Factoring by grouping is a method used to factor polynomials with four terms by grouping them into pairs. The aim is to simplify the polynomial into a product of binomials. Here's a simplified breakdown:
- First, you identify pairs of terms in the polynomial.
- Then, factor out the Greatest Common Factor (GCF) from each pair.
Coefficients and Constants
In the quadratic expression ax^2 + bx + c, 'coefficients' are these constants multiplying the variable terms.In our example 35y^2 + 34y + 8:
- a (coefficient of y^2) is 35
- b (coefficient of y) is 34
- c (constant term) is 8
Common Binomial Factor
A key step in factoring is finding a common binomial factor. This factor appears in each grouped segment of the expression. Using the previous example: ewline Once you have 7y(5y + 2) + 4(5y + 2), notice that (5y + 2) is found in both parts.This common factor can be factored out, or 'pulled out', of the expression to simplify it further:ewline (5y + 2)(7y + 4).The common binomial factor effectively simplifies the expression into a product of two binomials, making it easier to handle in equations or further operations.
