Algebra

Lesson

Let’s revisit some of the components of algebraic expressions that we’ve seen in earlier chapters by example.

Component | Algebraic expresison | Example |
---|---|---|

Variables | $x^2-x-6$x2−x−6 |
$x$x |

Terms | $x^2-x-6$x2−x−6 |
$x^2$x2, $-x$−x, and $-6$−6 |

Constant Terms | $x^2-x-6$x2−x−6 |
$-6$−6 |

Like Terms | $x^2-x-6+3x^2$x2−x−6+3x2 |
$x^2$x2 and $3x^2$3x2 |

Factors | $x^2-x-6$x2−x−6 |
$x-3$x−3,$x+2$x+2, $x^2-x-6$x2−x−6 and $1$1 |

Coefficients | $3x^2-x-6$3x2−x−6 |
$3$3 is a coefficient of $x^2$x2, $-1$−1 is a coefficient of $x$x |

Which of the options are factors of the expression below? Select all that apply.

$9\left(x+2\right)^2$9(

`x`+2)2$x$

`x`A$9$9

B$9\left(x+2\right)^2$9(

`x`+2)2C$2$2

D$x+2$

`x`+2E$\left(x+2\right)^2$(

`x`+2)2F$x$

`x`A$9$9

B$9\left(x+2\right)^2$9(

`x`+2)2C$2$2

D$x+2$

`x`+2E$\left(x+2\right)^2$(

`x`+2)2F

What is the numerical coefficient of the variable?

$-2.58t^4$−2.58`t`4

Consider the following expression.

$5x^3-3x^2-3x+10+9x^2$5`x`3−3`x`2−3`x`+10+9`x`2

Which of the following is like terms with $-3x^2$−3

`x`2?$10$10

A$9x^2$9

`x`2B$-3x$−3

`x`C$5x^3$5

`x`3D$10$10

A$9x^2$9

`x`2B$-3x$−3

`x`C$5x^3$5

`x`3D

We often want to substitute values for the variables in an algebraic expression. That way we can evaluate the expression to yield a numerical result. Let’s go through an example below.

Consider the algebraic expression $a^2+2ab-4c^2+\sqrt{a}$`a`2+2`a``b`−4`c`2+√`a`. Let's evaluate this expression when $a=4$`a`=4, $b=3$`b`=3 and $c=2$`c`=2.

Substituting $a=4$`a`=4, $b=3$`b`=3 and $c=2$`c`=2, the expression becomes:

$a^2+2ab-4c^2+\sqrt{a}$a2+2ab−4c2+√a |
$=$= | $\left(4\right)^2+2\times\left(4\right)\times\left(3\right)-4\left(2\right)^2+\sqrt{\left(4\right)}$(4)2+2×(4)×(3)−4(2)2+√(4) |

$=$= | $16+24-16+2$16+24−16+2 | |

$=$= | $26$26 |

Evaluate $\left(u+v\right)\left(w-y\right)$(`u`+`v`)(`w`−`y`) when $u=5$`u`=5, $v=8$`v`=8, $w=2$`w`=2 and $y=10$`y`=10.

Find the value of $\frac{x^2}{3}+\frac{y^3}{2}$`x`23+`y`32 when $x=-4$`x`=−4 and $y=3$`y`=3.

For $x=5$`x`=5 and $y=4$`y`=4,

Evaluate: $\sqrt{2x^2+4y+6}$√2`x`2+4`y`+6 correct to two decimal places.

When we have two expressions separated by an equal sign, we have an **equation**. In the case that our equation contains many variables, we can find the value of one variable by substituting values for the other variables.

Consider the equation of a parabola $y=ax^2+bx+c$`y`=`a``x`2+`b``x`+`c`. Find the value of $y$`y` when $a=1$`a`=1, $b=4$`b`=4, $c=4$`c`=4 and $x=2$`x`=2.

$y=ax^2+bx+c$y=ax2+bx+c |

$y=\left(1\right)\left(2\right)^2+\left(4\right)\times\left(2\right)+\left(4\right)$y=(1)(2)2+(4)×(2)+(4) |

$y=4+8+4$y=4+8+4 |

$y=16$y=16 |

Sometimes the variable of interest isn’t the subject of the equation. In this case, we will have to do some work to rearrange the equation so that the variable becomes the subject.

Consider the equation of a parabola again, $y=ax^2+bx+c$`y`=`a``x`2+`b``x`+`c`. Find the value of $x$`x` when $y=0$`y`=0, $a=1$`a`=1, $b=4$`b`=4 and $c=4$`c`=4.

$y=ax^2+bx+c$y=ax2+bx+c |

$0=\left(1\right)x^2+\left(4\right)x+\left(4\right)$0=(1)x2+(4)x+(4) |

$0=x^2+4x+4$0=x2+4x+4 |

$0=\left(x+2\right)^2$0=(x+2)2 |

$0=x+2$0=x+2 |

$x=-2$x=−2 |

Here we've used the fact that we can write $x^2+4x+4$`x`2+4`x`+4 as a perfect square, that is $\left(x+2\right)^2$(`x`+2)2.

Consider the equation $y=\left(bx+c\right)\left(dx+e\right)$`y`=(`b``x`+`c`)(`d``x`+`e`). Find the value of $y$`y` when $bx+c=-4$`b``x`+`c`=−4 and $dx+e=5$`d``x`+`e`=5.

$y=\left(bx+c\right)\left(dx+e\right)$y=(bx+c)(dx+e) |

$y=-4\times5$y=−4×5 |

$y=-20$y=−20 |

$x=-2$x=−2 |

Here we've used the fact that we can replace a whole algebraic expression with a given numerical value.

Some further examples are given below.

$P=2\left(a+b\right)$`P`=2(`a`+`b`) is the formula that describes the perimeter of the rectangle. Find $b$`b` if $a=2$`a`=2 and $P=22$`P`=22.

For $T=a+\left(b-1\right)d$`T`=`a`+(`b`−1)`d` find $d$`d` when $a=5$`a`=5, $b=2$`b`=2 and $T=8$`T`=8.

The thermal inertia ($I$`I`) of an object is given by $I=\sqrt{kpc}$`I`=√`k``p``c`, where $k$`k` is thermal conductivity, $p$`p` is the density and $c$`c` is the specific heat capacity.

Find the value of $c$

`c`in terms of $p$`p`when $I=4$`I`=4 and $k=9$`k`=9.