Problem 32 - Elementary Algebra

Question

Use the \(a c\) method to factor. Check the factoring. Identify any prime polynomials. $$ 3 u^{2}+14 u+8 $$

Step-by-Step Solution

Answer

The factored form is \( (3u + 2)(u + 4) \).

Step 1 - Multiply 'a' and 'c'

The given polynomial is in the form of \[ a u^{2} + b u + c \]where \( a = 3 \), \( b = 14 \), and \( c = 8 \). First, multiply 'a' and 'c'.\[(a \times c) = 3 \times 8 = 24 \].

Step 2 - Find two numbers that multiply to 'ac' and add to 'b'

Identify two numbers that multiply to 24 and add to 14. These numbers are 12 and 2, since \( 12 \times 2 = 24 \) and \( 12 + 2 = 14 \).

Step 3 - Rewrite the middle term using the two numbers

Rewrite the polynomial by splitting the middle term, 14u, into two terms using 12 and 2: \[ 3 u^2 + 12 u + 2 u + 8 \].

Step 4 - Factor by grouping

Group the terms in pairs and factor out the common factor from each pair: \[ 3u^2 + 12u + 2u + 8 \] can be written as \[ (3u^2 + 12u) + (2u + 8) \].Factor out the common factor from each binomial:\[ 3u(u + 4) + 2(u + 4) \].

Step 5 - Factor out the common binomial factor

Since both terms contain a common binomial factor \( (u + 4) \), factor it out: \[ (3u + 2)(u + 4) \].

Step 6 - Verify the factoring

Expand \( (3u + 2)(u + 4) \) to check: \[ 3u \times u + 3u \times 4 + 2 \times u + 2 \times 4 \] = \[ 3u^2 + 12u + 2u + 8 \] = \[ 3u^2 + 14u + 8 \]. The factorization is correct.

Key Concepts

ac method

The ac method is a technique used to factor trinomials of the form \[ au^2 + bu + c \]. It's a systematic way to find two numbers that help break down the trinomial into simpler factors. Start by multiplying the coefficients 'a' and 'c' from the trinomial.In the example of \[3u^2 + 14u + 8\], we have \(a = 3\) and \(c = 8\). When we multiply these, we get \[3 \times 8 = 24\].The next step is to find two numbers that multiply to this product (24) and sum to the middle coefficient 'b', which is 14. This strategic breakdown is crucial because it helps in splitting the middle term to use in the next steps.

polynomial factorization

Polynomial factorization is the process of breaking a polynomial into a product of simpler polynomials. The goal is to express the polynomial as a product of factors that, when multiplied together, give the original polynomial.For instance, with \[3u^2 + 14u + 8\], after using the ac method to find 12 and 2, we rewrite the polynomial by splitting the middle term. So, we have: \[3u^2 + 12u + 2u + 8\]. This step transforms the trinomial into a polynomial that can be factorized further using the grouping method.By breaking down the polynomial in parts, it becomes easier to see how these parts can recombine to form the original polynomial.

factoring by grouping

Factoring by grouping involves grouping terms to factor out the common elements within each group. This method works seamlessly after the ac method splits the middle term.In our example, we rewrite \[3u^2 + 14u + 8\] as \[3u^2 + 12u + 2u + 8\]. Next, group the terms into pairs: \[(3u^2 + 12u) + (2u + 8)\].Within each group, factor out the greatest common factor (GCF). For \[3u^2 + 12u\], factor out \(3u\): \[3u(u + 4)\]. For \[2u + 8\], factor out 2: \[2(u + 4)\]. After factoring, you will have: \[3u(u + 4) + 2(u + 4)\].Notice both groups have a common binomial factor \((u + 4)\). Now, we factor out \((u + 4)\) from both terms: \[ (3u + 2)(u + 4) \]. This completes the factorization.

prime polynomials

A polynomial is considered prime if it cannot be factored into polynomials with integer coefficients. To check if a polynomial is prime, try factoring it using various methods like ac method, grouping, or others. If none work, then the polynomial might be prime.In our example, after following the steps to factorize \[3u^2 + 14u + 8\], we end up with \[(3u + 2)(u + 4)\]. Since we successfully factorized it, \[3u^2 + 14u + 8\] is not a prime polynomial.Prime polynomials play a similar role to prime numbers—they are the building blocks of more complex polynomial expressions. Recognizing when a polynomial is prime is important for proper factorization and simplifying equations.