Factor 6×4 – 5×2 + 12×2 – 10 by grouping. What is the resulting expression? (6x + 5)(x2 – 2) (6x – 5)(x2 + 2) (6×2 + 5)(x2 – 2) (6×2 – 5)(x2 + 2)
The correct answer and explanation is:
We are given the expression: 6×4−5×2+12×2−106x^4 – 5x^2 + 12x^2 – 10
Step 1: Group the terms
Group the terms into two binomials: (6×4−5×2)+(12×2−10)(6x^4 – 5x^2) + (12x^2 – 10)
Step 2: Factor each group
Factor out the greatest common factor (GCF) from each group.
- From the first group 6×4−5x26x^4 – 5x^2, factor out x2x^2:
x2(6×2−5)x^2(6x^2 – 5)
- From the second group 12×2−1012x^2 – 10, factor out 2:
2(6×2−5)2(6x^2 – 5)
So now the expression becomes: x2(6×2−5)+2(6×2−5)x^2(6x^2 – 5) + 2(6x^2 – 5)
Step 3: Factor by grouping
Now that both terms share a common factor of (6×2−5)(6x^2 – 5), factor it out: (x2+2)(6×2−5)(x^2 + 2)(6x^2 – 5)
This is the fully factored expression.
Final Answer:
(6×2−5)(x2+2)(6x^2 – 5)(x^2 + 2)
Or rearranged to match the answer choices: (6×2−5)(x2+2)(6x^2 – 5)(x^2 + 2)
Correct Option:
(6x² – 5)(x² + 2)
Explanation (300 words):
Factoring by grouping is a method used when an expression has four terms. The goal is to rearrange and group terms so that you can factor out a common binomial from both groups. In the expression 6×4−5×2+12×2−106x^4 – 5x^2 + 12x^2 – 10, start by grouping the terms in pairs: (6×4−5×2)+(12×2−10)(6x^4 – 5x^2) + (12x^2 – 10)
From the first group, notice that both terms share a factor of x2x^2, so we factor it out: x2(6×2−5)x^2(6x^2 – 5)
In the second group, both terms share a factor of 2: 2(6×2−5)2(6x^2 – 5)
Now we have: x2(6×2−5)+2(6×2−5)x^2(6x^2 – 5) + 2(6x^2 – 5)
The binomial (6×2−5)(6x^2 – 5) is common to both terms, so we factor it out: (6×2−5)(x2+2)(6x^2 – 5)(x^2 + 2)
This is the factored form of the original expression. This method simplifies higher-order polynomials and is especially useful when direct factoring isn’t obvious. The key is recognizing common patterns and carefully factoring out the GCF from each group.
So, the correct answer is (6x² – 5)(x² + 2).