This section will look at some techniques for expanding and simplifying algebraic expressions. In the first instance, we will be looking at the distributive law, which is defined mathematically as:
$a.(b+c)=a.b+a.c$
We can simply demonstrate that this is true, though proving it is, like so much mathematics, beyond the scope of this section.
$2(5+7) = 2 \times 12= 24$
$2(5+7) = 2 \times 5 + 2 \times 7 = 24$
This will lead us into factorising which is the same process, but backwards. This is what we can an inverse operation.
Worked example 1: Mutiplying a bracket by a single term
- Notes
Multiply out and simplify the expression $$4(p+3)$$
$$\begin{aligned}&4(p+3) \\& = (4 \times p) + (4 \times 3) \\& = 4p + 12 \end{aligned}$$
In the image here, you can see how we take the $4$ from outside the bracket and multiply it by each term inside the brackets.
$4 \times p = 4p$ and $4 \times 3 = 12$. These terms contain no like terms, so we just write down the answer.
$$4p+12$$
Worked example 2: Multiplying a bracket by a more complex term
- Notes
Expand and simplify the expression $3x^3(4x+2y)$
$\begin{aligned}&3x^3(4x+2y) \\ &=12x^4+6x^3y \end{aligned}$
The principle here is identical to the previous example. The steps likewise, are the same. Each term in the brackets $4x$ and $2y$ must be multiplied by $3x^3$.
$3x^3 \times 4x$
$= 12x^3 \times x^1$
$=12x^4$. Remember the first law of indices
($x^p \times x^q = x^{p+q}$)$12x^4+6x^3y$
Worked example 3: Multiplying out two sets of brackets (Parrot's Beak Method)
- Notes
Expand and simplify $(x+3)(x+2)$
The image to the left shows the Parrot's Beak Method. This is identical to the famous FOIL method, but marginally more entertaining. My parrot usually (as in the image) looks more like a pigeon, but I hope it makes the point.
F First by first
O Outside pair
I Inside pair
L Last by last
Another thing to notice and file away for future reference is that the constant in the 2nd term is $5$ which is $3+2$ and the constant in the 3rd term is $6$, which is $3 \times 2$. This observation is going to be very helpful when we start on factorising algebraic expressions, so hold the thought.
- $x^2+5x+6$
Exercise 1: Multiplying a +ve number by an expression in brackets
- Instructions:Expand and simplify the following expressions:
- Q1\[2 (x + 1)\]
$2x + 2$ - Q2\[3 (b + 1 )\]
$3b + 3$
- Q3\[5 (q + 2 )\]
$5q + 10$
- Q4\[4 (y + 3)\]
$4y + 12$
- Q5\[2 (a + 10)\]
$2a + 20$
- Q6\[7 (b + 3)\]
$7b + 21$
- Q7\[8 ( d - 1 )\]
$8d - 8$
- Q8\[7 (c - 5 ) \]
$7c - 35$
- Q9\[6 (e - 3 )\]
$6e - 18$
- Q10\[7 (g - 2 )\]
$7g - 14$
Exercise 2: Multiplying a bracket by a negative expression
- Instructions:Expand and simplify the following expressions:
- Q1\[-2(f+2 )\]
$-2f-4$
- Q2\[-3(n+3 )\]
$-3n-9$
- Q3\[-4(4x+1 )\]
$-16x-4$
- Q4\[-1(2p+1 )\]
$-2p-1$
- Q5\[-(a+3 )\]
$-a-3$
- Q6\[-(4j-3 )\]
$-4j+3$
- Q7\[-(-k-1 )\]
$k+1$
- Q8\[-3(5-r )\]
$-15+3r$
- Q9\[-10(4m+1 )\]
$-40m-10$
- Q10\[-5(-p+4 )\]
$5p-20$
Exercise 3: Mutiplying a bracket by a single term involving powers
- Instructions:Multiply out the following expressions.
- Q1\[e (e-1 )\]
$e^2-e$
- Q2\[k (k+2 )\]
$k^2+2k$
- Q3\[q (q-1 )\]
$q^2-q$
- Q4\[a (2a+b )\]
$2a^2+ab$
- Q5\[h (3h+2 )\]
$3h^2+2h$
- Q6\[m (3m-4 )\]
$3m^2-4m$
- Q7\[c (4d-c )\]
$4cd-c^2$
- Q8\[2g (3g+4 )\]
$6g^2+8g$
- Q9\[5r(5r-6)\]
$25r^2-30r$
- Q10\[p (-p-u )\]
$-p^2-pu$
Exercise 4: More complicated examples
- Instructions:Multiply out the following expressions and simplify your answers if possible.
- Q1\[t^2 (4t^2-3t )\]
$4t^4-3t^3$
- Q2\[-x^2 (3x^2+8x )\]
$-3x^4-8x^3$
- Q3\[x^2y (8x+y )\]
$8x^3y+x^2y^2$
- Q4\[-n^2 (2n-3 )\]
$-2n^3+3n^2$
- Q5\[-p^2 (3p+4p^2 )\]
$-3p^3-4p^4$
- Q6\[2 (b+1 )+5 (b+2 )\]
$7b+12$
- Q7\[2 (w+2 )+3 (w+3 )\]
$5w+13$
- Q8\[3 (s+5 )+2 (s+3 )\]
$5s+21$
- Q9\[4 (2m+3 )+5 (3m+2 )\]
$23m+22$
- Q10\[2 (2r-3 )+4 (3r-2 )\]
$16r-14$
- Q11\[6 (3q-5 )+2 (4q-3 )\]
$26q-36$
- Q12\[3 (3b+4 )-2 (2b+3 )\]
$5b+6$
- Q13\[5 (4p+7 )-3 (3p+4 )\]
$11p+23$
- Q14\[6 (2j-5 )- (4j+1 )\]
$8j-31$
- Q15\[5 (4q+5 )-3 ( 5q-1)\]
$5q+22$
- Q16\[5a (a-3 )-2a (2a-5 )\]
$a^2-5a$
- Q17\[7b (2b-3c )-3c (2c-4b )\]
$14b^2-9bc-6c^2$
- Q18\[-d (-e-9d^2 )\]
$de+9d^3$
Exercise 5: Mixed exercise
- Instructions:Expand and simplify the following
- Q1\[5 (x+3 )\]
$5x+15$
- Q2\[3(b-1)\]
$3b-3$
- Q3\[7(q-3)\]
$7q-21$
- Q4\[4(y-3)\]
$4y-12$
- Q5\[5(a+b)\]
$5a+5b$
- Q6\[6(b-4)\]
$6b-24$
- Q7\[d(d-1)\]
$d^2-d$
- Q8\[c(2+c)\]
$2c+c^2$
- Q9\[e(e-3)\]
$e^2-3e$
- Q10\[2p(p+5)\]
$2p^2+10p$
- Q11\[5y(y+2)\]
$5y^2+10y$
- Q12\[-(3x+4)\]
$-3x-4$
- Q13\[-(5r-6)\]
$-5r+6$
- Q14\[-(10-t)\]
$-10+t$ or $t-10$
- Q15\[7(a+3)+4(a-3)\]
$11a+9$
- Q16\[8(6p-5)-3(2p-5)\]
$42p-25$
- Q17\[4f(2f+5)\]
$8f^2+20f$
- Q18\[5w(w^2+5)\]
$5w^3+25w$
- Q19\[11t(3-5t)\]
$33t-55t^2$
- Q20\[6q(4-q)\]
$24q-6q^2$