Expanding a Linear & a Quadratic (WJEC GCSE Maths & Numeracy (Double Award): Higher): Revision Note

Expanding a Linear & a Quadratic

How do I expand a linear expression and a quadratic expression?

• The same principles apply as to when you expand any combination of expressions or brackets

• Every term in the first bracket must be multiplied by every term in the second bracket

  • Expanding$(3x^2+4x-5)(3x+2)$ requires 6 multiplications in total

• When you multiply a quadratic by a linear term, as above, you will get a cubic $\left(x^3\right)$

• As there are lots of terms to keep track of, you may prefer to use a grid

• To expand $(3x^2+4x-5)(3x+2)$,

  • Write out the brackets as row and column headings of a grid

  • They can be in either direction

  • Remember to write the appropriate sign in front of each term

• For each cell in the grid, multiply the term in the row heading by the term in the column heading

  • Remember that $x^2\times x=x^3$

• Add together all the terms inside the grid to get the answer

  • $9x^3+6x^2+12x^2+8x-15x-10$

• Collect like terms

  • $9x^3+18x^2-7x-10$

What if there are three brackets?

• You may have to expand three brackets like $\left(2a+3\right)\left(7a-2\right)\left(a+4\right)$

• You know that the first two brackets $\left(2a+3\right)\left(7a-2\right)$ expand to form a quadratic

  • Therefore $\left(2a+3\right)\left(7a-2\right)\left(a+4\right)$ will expand to form a cubic

• To expand three brackets,

  • Expand the first two brackets using your preferred method and simplify

    • $\left(2a+3\right)\left(7a-2\right)=14a^2-4a+21a-6=14a^2+17a-6$

  • Multiply the result of this by the remaining bracket

    • $\left(14a^2+17a-6\right)\left(a+4\right)=14a^3+56a^2+17a^2+68a-6a-24$

  • Simplify your answer

    • $14a^3+73a^2+62a-24$