Algebraic Expressions

Substitution #

Swap the letter for a number and work it out.

Substituting means putting a number where the letter stands

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The expression 3n just means 3 lots of whatever n is.

Let n be 4 and the whole expression becomes 12.

The 4 goes in the box — and only in the box. The 2 waits outside it.

Like Terms #

Only the same letter part can be added.

Only terms with exactly the same letter part can be added together

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3 of something and 2 more of the same thing makes 5 of it: 3a + 2a = 5a.

3a and 2a are like terms, so they join into 5a. A 2b would stay apart.

Expanding Brackets #

Outside multiplies everything inside.

Everything outside a bracket multiplies everything inside it

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A rectangle 3 by 5 has 3 × 5 = 15 squares inside it.

Cut the width into 3 and 2 and the area splits the same way: 3 × (3 + 2) = 3 × 3 + 3 × 2.

The sides can be letters: a(x + b) is the ax strip plus the ab strip.

Factoring by Common Factor #

Pull out what both parts share.

Factoring is putting the bracket back by pulling out what both share

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4 × 3 plus 4 × 4, and both lots share the 4.

Take the shared 4 outside and the rest goes into the bracket.

Factoring by Grouping #

Four terms, taken two at a time.

Four terms with no common factor can still split into two pairs that share one

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No single factor divides all four, so group them two at a time — each pair shares one.

Both pairs leave the same bracket behind: y + 3. That is the whole method.

Two terms, both holding y + 3, so take it out: (y + 3)(x + 2).

In letters it is the same three moves, and the matching bracket is a + b.

Expanding Two Brackets #

Four products, one from each pair of terms.

Every term in one bracket multiplies every term in the other, making four pieces

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Draw a rectangle x + 3 wide and x + 2 tall. Its whole area is that product.

Both cuts split it into 4 pieces: x × x, x × 3, 2 × x, 2 × 3.

The two middle pieces are both lots of x, so 3x and 2x join into 5x.

Take only the corner pieces and 5x is left out — that is the usual slip.

Factoring Quadratics #

Find the pair that multiply and add right.

Factoring a quadratic means finding the pair that multiply and add correctly

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$x^2+5x+6$x² + 5x + 6 came from a rectangle. Which split of its sides makes 5x and 6?

2 × 3 = 6 and 2 + 3 = 5, so 2 and 3 are the numbers.

Factoring Quadratics with a Leading Coefficient #

The pair hunt moves to a times c.

A leading coefficient moves the pair hunt to a times c, then the middle term splits

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A 2 sits in front now, so the pair must multiply to 2 × 3 = 6, not to 3 alone.

Hunt the pair for 6 that adds to 7: 1 and 6. Those split the middle term.

Split 7x as x + 6x. Nothing changed — the two pieces still add back to 7x.

Factor each half and the same bracket appears twice — pull it out, and done.

The rectangle checks it: the four pieces add back to $2x^2+7x+3$2x² + 7x + 3.

Algebraic Identities #

The expansions that keep coming back.

A few expansions come up so often they are worth knowing by sight

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A square of side a + b holds $a^2$a², two ab strips and $b^2$b² — so the middle is 2ab.

Each piece left keeps an (x − b) side, so the three together are (x − b)(x + b).

The Sum and Difference of Two Cubes #

A linear bracket times a quadratic one.

A sum or difference of two cubes factors into a linear bracket times a quadratic one

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Multiply it out and the middle terms cancel, leaving only the two cubes.

Flip the sign in the linear bracket and the middle sign in the quadratic one for the sum.

Write the number as a cube first, then a and b can be read straight off.

For a sum: 27 is $3^3$3³, so $x^3+27$x³ + 27 factors as $\left(x+3\right)\left(x^2-3x+9\right)$(x + 3)(x² − 3x + 9).

Finding Unknown Coefficients by Matching #

Two names for one expression fix every letter.

Two expressions equal for every x must agree part by part, which pins the unknowns

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Expand the left side. One expression is now written out in two ways.

Equal for every x is strong: each matching part must be equal on its own.

One statement became two, and the pair adding to 7 and timesing to 12 is 3 and 4.

Same move on p(x + 3) = 4x + q: the x terms give p, and p then gives q.

Algebraic Fractions #

Cancel only a factor of the whole thing.

An algebraic fraction cancels only what is a factor of the whole top and bottom

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3x over 6x: the x is a factor of both, so it goes.

What is left is one half, and only a whole factor could be canceled.

Adding and Multiplying Algebraic Fractions #

Common bottoms to add, straight across to multiply.

Algebraic fractions add over a common denominator and multiply straight across

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Same rule as unlike fractions: rewrite both over 6, and then the tops can add.

Cut x into sixths: half of it is three sixths, a third is two — five in all.

Multiplying is easier: multiply the numerators, then the denominators — $\frac{x^2}{6}$x²/6.

The slip to dodge: adding the bottoms. No fifths ever appear — the parts are sixths.

Recognizing Equivalent Expressions #

Equal at every x, or a near-miss that is not.

Equivalent means equal at every x — one substitution can refute, none can prove

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Expanding never changes the value: 3(x + 2) and 3x + 6 agree at every x.

3x + 2 is the near-miss — the 2 never met the × 3. Try x = 1: 9 against 5.

Squaring a sum grows a middle term: ${\left(x+3\right)}^2$(x + 3)² is $x^2+6x+9$x² + 6x + 9, never $x^2+9$x² + 9.

Agreeing at one x proves nothing: $x^2$x² and 2x meet at x = 2. Algebra covers every x.