Problem 39
Question
Use the \(a c\) method to factor. Check the factoring. Identify any prime polynomials. $$ 4 x^{2}+20 x+25 $$
Step-by-Step Solution
Verified Answer
The factored form is \text(short_answer): [(2x + 5)^2\]. It is not a prime polynomial.
1Step 1: Identify a, b, and c
Given the quadratic equation of the form \[ax^2 + bx + c\], identify the coefficients. Here, \[a = 4\], \[b = 20\], and \[c = 25\].
2Step 2: Calculate ac
Multiply the coefficients \[a\] and \[c\] to get \[ac\]. For this equation, \[ac = 4 \times 25 = 100\].
3Step 3: Find two numbers that multiply to ac and add to b
Find two numbers that multiply to \[ac = 100\] and add to \[b = 20\]. The numbers 10 and 10 satisfy this requirement as \[10 \times 10 = 100\] and \[10 + 10 = 20\].
4Step 4: Rewrite the middle term using the two numbers
Rewrite the quadratic expression by splitting the middle term \[20x\] into \[10x + 10x\]. Now, the equation becomes \[4x^2 + 10x + 10x + 25\].
5Step 5: Factor by grouping
Group the terms to factor them: \[(4x^2 + 10x) + (10x + 25)\].
6Step 6: Factor out the greatest common factor (GCF) from each group
In the first group \[4x^2 + 10x\], factor out \[2x\] to get \[2x(2x + 5)\]. In the second group, \[10x + 25\], factor out \[5\] to get \[5(2x + 5)\]. Thus, the equation becomes \[2x(2x + 5) + 5(2x + 5)\].
7Step 7: Factor out the common binomial factor
Notice that \[2x + 5\] is a common factor. Factor \[(2x + 5)\] out of each term to get \[(2x + 5)(2x + 5)\] or \[(2x + 5)^2\].
8Step 8: Check the factoring
Expand \[(2x + 5)^2\] to verify the original polynomial: \[(2x + 5)(2x + 5) = 4x^2 + 10x + 10x + 25 = 4x^2 + 20x + 25\]. This matches the original polynomial.
9Step 9: Identify the polynomial type
Since the polynomial factors into \[(2x + 5)^2\], it is not a prime polynomial.
Key Concepts
ac methodprime polynomialsgreatest common factor
ac method
The ac method is a systematic way to factor quadratic equations, especially when the leading coefficient is not 1. To use this method, follow these steps:
First, identify the coefficients of the quadratic equation in the form \(ax^2 + bx + c\). These are \(a\), \(b\), and \(c\). For example, in the equation \(4x^2 + 20x + 25\), \(a = 4\), \(b = 20\), and \(c = 25\).
Next, calculate the product of \(a\) and \(c\). In this case, \(4 \times 25 = 100\), so \(ac = 100\).
Then, find two numbers that multiply to \(ac\) and add to \(b\). Here, both numbers are \(10\), because \(10 \times 10 = 100\) and \(10 + 10 = 20\).
Now, rewrite the middle term using these two numbers. Instead of \(20x\), write \(10x + 10x\), so the equation becomes \(4x^2 + 10x + 10x + 25\).
Group the terms: \((4x^2 + 10x) + (10x + 25)\).
Factor out the greatest common factor from each group. From \((4x^2 + 10x)\), factor out \(2x\) to get \(2x(2x + 5)\). From \((10x + 25)\), factor out \(5\) to get \(5(2x + 5)\).
Notice that \((2x + 5)\) is a common factor. Factor it out to get \((2x + 5)(2x + 5)\) or \((2x + 5)^2\).
Lastly, always check your work by expanding the factors to ensure they match the original polynomial.
First, identify the coefficients of the quadratic equation in the form \(ax^2 + bx + c\). These are \(a\), \(b\), and \(c\). For example, in the equation \(4x^2 + 20x + 25\), \(a = 4\), \(b = 20\), and \(c = 25\).
Next, calculate the product of \(a\) and \(c\). In this case, \(4 \times 25 = 100\), so \(ac = 100\).
Then, find two numbers that multiply to \(ac\) and add to \(b\). Here, both numbers are \(10\), because \(10 \times 10 = 100\) and \(10 + 10 = 20\).
Now, rewrite the middle term using these two numbers. Instead of \(20x\), write \(10x + 10x\), so the equation becomes \(4x^2 + 10x + 10x + 25\).
Group the terms: \((4x^2 + 10x) + (10x + 25)\).
Factor out the greatest common factor from each group. From \((4x^2 + 10x)\), factor out \(2x\) to get \(2x(2x + 5)\). From \((10x + 25)\), factor out \(5\) to get \(5(2x + 5)\).
Notice that \((2x + 5)\) is a common factor. Factor it out to get \((2x + 5)(2x + 5)\) or \((2x + 5)^2\).
Lastly, always check your work by expanding the factors to ensure they match the original polynomial.
prime polynomials
A prime polynomial is a polynomial that cannot be factored further over the integers. This means it doesn't break down into simpler polynomials multiplied together.
A polynomial is considered prime if it has no non-trivial factors other than itself and 1. For example, \(x^2 + 1\) is prime because it cannot be factored using real numbers.
In the given exercise, we factorized the quadratic equation \(4x^2 + 20x + 25\) successfully into \((2x + 5)^2\). Since it factors, it is not a prime polynomial.
If, however, a polynomial cannot be factored into simpler polynomials with integer coefficients, it's identified as prime. Remember, a prime polynomial is like a prime number. It stands alone and cannot be split into the product of other polynomials without involving fractions or complex numbers.
A polynomial is considered prime if it has no non-trivial factors other than itself and 1. For example, \(x^2 + 1\) is prime because it cannot be factored using real numbers.
In the given exercise, we factorized the quadratic equation \(4x^2 + 20x + 25\) successfully into \((2x + 5)^2\). Since it factors, it is not a prime polynomial.
If, however, a polynomial cannot be factored into simpler polynomials with integer coefficients, it's identified as prime. Remember, a prime polynomial is like a prime number. It stands alone and cannot be split into the product of other polynomials without involving fractions or complex numbers.
greatest common factor
The Greatest Common Factor (GCF) is the largest number or expression that divides two or more expressions without any remainder. It is a key step in factoring polynomials.
To find the GCF, look at each term in the polynomial. Identify the greatest factor that is common to all terms.
For example, in \((4x^2 + 10x)\), the common factor is \(2x\). It's the largest expression that divides both \((4x^2)\) and \((10x)\) without a remainder.
In the exercise, the GCF is used in the grouping step. We factored out \(2x\) from \((4x^2 + 10x)\) and \((5)\) from \((10x + 25)\). Recognizing and extracting the GCF simplifies the polynomial and helps us factor it efficiently.
Always ensure you check terms for their GCF before moving ahead in factoring as it simplifies your work and ensures accuracy.
To find the GCF, look at each term in the polynomial. Identify the greatest factor that is common to all terms.
For example, in \((4x^2 + 10x)\), the common factor is \(2x\). It's the largest expression that divides both \((4x^2)\) and \((10x)\) without a remainder.
In the exercise, the GCF is used in the grouping step. We factored out \(2x\) from \((4x^2 + 10x)\) and \((5)\) from \((10x + 25)\). Recognizing and extracting the GCF simplifies the polynomial and helps us factor it efficiently.
Always ensure you check terms for their GCF before moving ahead in factoring as it simplifies your work and ensures accuracy.
Other exercises in this chapter
Problem 39
Factor completely. Identify any prime polynomials. $$ 6 x^{2}-6 y^{2} $$
View solution Problem 39
Use a pattern to factor. Check. Identify any prime polynomials. $$ h^{3}-27 $$
View solution Problem 39
(a) factor out the greatest common factor. Identify any prime polynomials. (b) check. $$ p^{4}-p^{3}-p^{2}-p $$
View solution Problem 40
(a) solve. (b) check. $$ n^{2}-n-30=0 $$
View solution