Mathematics 10 — Exercise Review · Q2 (iii)

Full question

For the following functions, find fg.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$\left(25-x^2\right)\times \left(x+5\right)$$

2. Distribute each bracket product: each term multiplies every term in the other bracket.

   $$-x^3-5\times x^2+25\times x+125$$

3. Write EVERY distributed product before collecting terms.

   $$25\times 5+25\times x+5\times \left(-x^2\right)+x\times \left(-x^2\right)$$

4. Calculate this individual product, including its sign and powers.

   $$25\times 5=125$$

5. Calculate this individual product, including its sign and powers.

   $$25\times x=25\times x$$

6. Calculate this individual product, including its sign and powers.

   $$5\times \left(-x^2\right)=-5\times x^2$$

7. Calculate this individual product, including its sign and powers.

   $$x\times \left(-x^2\right)=-x^3$$

8. These terms have the same variable part. Add ONLY their coefficients.

   $$125=125$$

9. These terms have the same variable part. Add ONLY their coefficients.

   $$25=25$$

10. These terms have the same variable part. Add ONLY their coefficients.

    $$-5=-5$$

11. These terms have the same variable part. Add ONLY their coefficients.

    $$-1=-1$$

12. Put the simplified groups together.

    $$125+25\times x-5\times x^2-x^3$$

13. Collect terms with the same power and simplify their coefficients.

    $$-x^3-5\times x^2+25\times x+125$$

Answer

The final answer is shown below.