Expand and simplify the expression:
$\big(3p+4 \big)\big(4p-1 \big)$
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Multiplying brackets $ \big(x \pm a \big)\big(x \pm b \big)$
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Multiplying brackets $ \big(mx \pm a \big)\big(nx \pm b \big)$
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Factorizing expressions
Work slowly and carefully
When multiplying out more complex brackets or multiplying out three of them, it is crucial to work systematically, slowly & carefully. The faster you go, the more mistakes you are likely to make.
Worked example 1: Multiplying two brackets
- Notes
\[\begin{array}{l} (3p + 4)(4p - 1)\\ = 12{p^2} - 3p + 16p - 4\\ = 12{p^2} + 13p - 4 \end{array}\]Remember not to rush.
Always use the method you prefer. After completing each term, check that the
- Numbers are correct
- Signs are correct.
- The answer is fully simplified
$12p^2+13p-4$
Worked example 2: Multiplying two brackets and a term
- Notes
Expand and simplify $v \big(v+2\big)\big(v-2\big)$
- \[\begin{array}{l} v(v + 2)(v - 2)\\ = v(v^2 - 2v + 2v - 4)\\ = v(v^2 - 4)\\ = v^3 - 4v \end{array}\]
Always leave the single term until last. If you multiply by the $v$ first, then everything becomes more difficult.
Remember that you should write down the whole expression on each line. This means that you should write down the $v($ and then concentrate on multiplying out the quadratic, which should go in a bracket. Simplify this expression, and then (and only then) multiply the bracket by the $v$.
- $v^3-4v$
Worked example 3: Multiplying three brackets
- Notes
Expand and simplify the expression:
$\big(2a-3\big)\big(a+4\big)\big(3a-5\big)$- \[\begin{array}{l} (2a - 3)(a + 4)(3a - 5)\\ = (2{a^2} + 4a - 3a - 12)(3a - 5)\\ = (2{a^2} + a - 12)(3a - 5)\\ = 6{a^3} - 10{a^2} + 3{a^2} - 5a - 36a + 60\\ = 6{a^3} - 7{a^2} - 41a + 60 \end{array}\]
It doesn't matter which of the three brackets you start with. I have multiplied out the first two brackets first. Note that the third bracket is written at the end of each line, where we do nothing with it until line 4 where we multiply the quadratic by the third bracket and then finish off with some simplification.
Please also note that line 3 exists only to simplify the expression above. It can be tempting to leave all simplification until the end; this is generally not a great idea.
- \[6{a^3} - 7{a^2} - 41a + 60\]
Exercise 1: Both signs the same
- Instructions:
Expand and simplify the following expressions:
- Q1\[(2f+1 )(f+1 )\]
$2f^2+3f+1$ - Q2\[(4m+3 )(m+1 )\]
$4m^2+7m+3$
- Q3\[(2w+5 )(w+4 )\]
$2w^2+13w+20$
- Q4\[(2n+3 )^2\]
$4n^2+12n+9$
- Q5\[(3z+2 )(2z+3 )\]
$6z^2+13z+6$
- Q6\[(3r-4 )(r-1 )\]
$3r^2-7r+4$
- Q7\[(2s-3 )(2s-4 )\]
$4s^2-14s+12$
- Q8\[(4d-3 )^2\]
$16d^2-24d+9$
- Q9\[(3r-4 )(r-1 )\]
$3r^2-7r+4$
- Q10\[(2s-3 )(2s-4 )\]
$4s^2-14s+12$
Exercise 2: Different signs
- Instructions:Expand and simplify the following expressions:
- Q1\[(2x-5 )(3x+2 )\]
$6x^2-11x-10$
- Q2\[(2h-3 )(2h+3 )\]
$2h^2-9$
- Q3\[(5-3j )(5+3j )\]
$25-9j^2$
- Q4\[(2+3c )(1-2c )\]
$2-c-6c^2$
- Q5\[(3a-1 )(a+2 )\]
$3a^2+5a-2$
- Q6\[(4-u )(4+u )\]
$16-u^2$
- Q7\[(3+2y )(4y-1 )\]
$8y^2+10y-3$
- Q8\[(2-3p )(p+2 )\]
$-3p^2-4p+4$
- Q9\[(3q+2v )(3q-2v )\]
$9q^2-4v^2$
- Q10\[(ax+b )(ax-b )\]
$a^2x^2-b^2$
Exercise 3: Multiplying three linear factors
- Instructions:
Expand and simplify the following expressions:
When multiplying a term by two brackets, multiply the brackets first, and then multiply by the term.
- Q1\[x(x+1)(x+2)\]
$x^3+3x^2+2x$
- Q2\[p(p+1)(p+3)\]
$p^3+4p^2+3p$
- Q3\[m(m-2)(m+5)\]
$m^3+3m^2-10m$
- Q4\[r(r-7)(r+3)\]
$r^3-4r^2-21r$
- Q5\[4f(f-2)(f-4)\]
$4f^3-24f^2+32f$
- Q6\[5g(g-2)(g+1)\]
$5g^3-5g^2-10g$
- Q7\[(a+1)^3\]
$a^3+3a^2+3a+1$
- Q8\[(r+1)^2(r+2)\]
$r^3+4r^2+5r+2$
- Q9\[(n+2)(n+3)(n+4)\]
$n^3+9n^2+26n+24$
- Q10\[-6y(4y-1)(3y+2 )\]
$-72y^3-30y^2+12y$