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Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus

Factoring a Trinomial: Multiply to c, Add to b

  • Grade 10
  • Foundations of Mathematics and Pre Calculus
  • 2 min 20 s
  • British Columbia
  • BC10-MA-32

To factor x² + 7x + 12, the video looks for two numbers that multiply to 12 and add to 7, which gives 3 and 4. It rebuilds the algebra tiles into one rectangle x + 4 wide and x + 3 tall, then multiplies out (x + 3)(x + 4) to check. It repeats the rule with signs to get x² − 2x − 15 = (x − 5)(x + 3), shows how swapping the two rules gives the wrong answer, and ends on the one-line recap.

Concept: Factoring a trinomial: two numbers that multiply to c and add to b

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In one sentence

The question this lesson answers

How do we factor the trinomial x squared plus 7 x plus 12? A trinomial has three terms.

Here b is 7, the number multiplying x, called the coefficient. c is 12, the number on its own, called the constant term.

Transcript

What the lesson says

How do we factor the trinomial x squared plus 7 x plus 12? A trinomial has three terms.

Here b is 7, the number multiplying x, called the coefficient. c is 12, the number on its own, called the constant term.

Factoring means writing it as a product of two brackets.

We need two numbers that multiply to 12 and add to 7.

Try the positive factor pairs of 12: 1 and 12 add to 13, 2 and 6 add to 8, 3 and 4 add to 7.

So it factors as x plus 3, times x plus 4.

To see why, draw each term as an area: an x by x square, 7 strips of x by 1, and 12 unit squares.

Put the square in a corner, 4 strips beside it and 3 strips below it.

The 12 unit squares fill the corner, making a rectangle x plus 4 wide and x plus 3 tall.

Check: x squared, plus 4 x, plus 3 x, plus 12, makes x squared plus 7 x plus 12.

Now with signs: x squared minus 2 x minus 15. Multiply to negative 15, add to negative 2.

One number must be negative. 5 and negative 3 add to 2, but negative 5 and 3 add to negative 2.

So it factors as x minus 5, times x plus 3.

Check: x squared, plus 3 x, minus 5 x, minus 15, makes x squared minus 2 x minus 15.

A common mistake is to swap the rules: add to c and multiply to b.

6 and 6 add to 12, but x plus 6, times x plus 6, is x squared plus 12 x plus 36, not our trinomial.

To factor x squared plus b x plus c, find two numbers that multiply to c and add to b.

Curriculum

Where this fits

British Columbia, Grade 10

Foundations of Mathematics and Pre Calculus 10, Algebra: polynomial factoringBC10-MA-32

Previous: Factoring Out the Greatest Common Factor

The outcomes and strands follow the British Columbia curriculum.

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