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One of the key skills in algebra, especially at A Level, is the ability to expand brackets involving polynomials. Let’s take a closer look at the following problem:
Expand the following brackets:
(2x^2 + 6x + 4)(3x^2 - x - 3)
This expression requires us to multiply two trinomials. The process may seem daunting at first, but with a systematic approach, it becomes much more manageable.
Step 1: Multiply Each Term in the First Bracket by Each Term in the Second
To expand, we take every term in the first bracket and multiply it by every term in the second bracket:
2x^2 \times 3x^2 = 6x^4 2x^2 \times (-x) = -2x^3 2x^2 \times (-3) = -6x^2 6x \times 3x^2 = 18x^3 6x \times (-x) = -6x^2 6x \times (-3) = -18x 4 \times 3x^2 = 12x^2 4 \times (-x) = -4x 4 \times (-3) = -12Step 2: List All the Terms
After multiplying out, we get:
6x^4 - 2x^3 - 6x^2 + 18x^3 - 6x^2 - 18x + 12x^2 - 4x - 12
Step 3: Combine Like Terms
Now, group and add the like terms:
x^4 terms: 6x^4
x^3 terms: -2x^3 + 18x^3 = 16x^3
x^2 terms: -6x^2 - 6x^2 + 12x^2 = 0x^2
x terms: -18x - 4x = -22x
Constant: -12
So the expanded expression is:
\boxed{6x^4 + 16x^3 - 22x - 12}
Why Is This Important?
Expanding brackets like this is not just an exercise in algebraic manipulation—it’s a foundation for many other areas in mathematics, including calculus, solving equations, and simplifying complex expressions. Mastery of these techniques is crucial for success at A Level and beyond.
If you ever feel unsure about these kinds of problems, seeking out Online Maths Tuition for A Level can give you the support and confidence you need. With regular practice and expert guidance, expanding even the trickiest brackets will soon feel straightforward.
Understanding how to expand and simplify polynomial expressions is a vital skill, and with a methodical approach, you’ll find yourself solving these with ease.
