O Level, A Level Math Solutions: New Elementary Mathematics ... - Free Printable

You've uploaded an image containing Problem 6 from Chapter 2, Exercise 2.3, Page 33, which asks to factorize several algebraic expressions. The problem includes parts (a), (b), (c), (j), (k), and (m), with solutions already provided.

Let me go through each part step by step, verify the factorizations, and explain the reasoning behind each solution. This will help you understand how these expressions are factored.

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(a) $ 25a^2 - 16b^2 $

Step-by-step:

We recognize this as a difference of squares:
$$
25a^2 - 16b^2 = (5a)^2 - (4b)^2
$$

Using the identity:
$$
A^2 - B^2 = (A + B)(A - B)
$$

So:
$$
= (5a + 4b)(5a - 4b)
$$

✔ Correct.

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(b) $ 9m^2 + 12mn + 4n^2 $

This is a perfect square trinomial.

Check:
- $ 9m^2 = (3m)^2 $
- $ 4n^2 = (2n)^2 $
- Middle term: $ 2 \cdot 3m \cdot 2n = 12mn $ → matches!

So:
$$
= (3m + 2n)^2 = (3m + 2n)(3m + 2n)
$$

✔ Correct.

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(c) $ 16p^2 - 24p + 9 $

Again, check if it's a perfect square trinomial.

- $ 16p^2 = (4p)^2 $
- $ 9 = 3^2 $
- Middle term: $ 2 \cdot 4p \cdot 3 = 24p $ → but we have –24p, so sign matters.

So:
$$
= (4p)^2 - 2 \cdot 4p \cdot 3 + 3^2 = (4p - 3)^2
$$

Thus:
$$
= (4p - 3)(4p - 3)
$$

✔ Correct.

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(j) $ (x + 1)^2 - 9 $

This is a difference of squares again:
$$
= (x + 1)^2 - 3^2
$$

Use:
$$
A^2 - B^2 = (A + B)(A - B)
$$

So:
$$
= [(x + 1) + 3][(x + 1) - 3] = (x + 4)(x - 2)
$$

✔ Correct.

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(k) $ (x - 1)^2 - (x + 1)^2 $

This is a difference of squares in disguise:
$$
= A^2 - B^2 = (A + B)(A - B)
$$
where $ A = x - 1 $, $ B = x + 1 $

So:
$$
= [(x - 1) + (x + 1)][(x - 1) - (x + 1)]
$$

Simplify:
- First bracket: $ x - 1 + x + 1 = 2x $
- Second bracket: $ x - 1 - x - 1 = -2 $

So:
$$
= (2x)(-2) = -4x
$$

✔ Correct.

Note: The final answer is $-4x$, which is fully simplified.

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(m) $ x^2 + yz + xy + xz $

Group terms strategically:

$$
= x^2 + xy + xz + yz
$$

Now group:
$$
= (x^2 + xy) + (xz + yz)
$$

Factor each group:
- $ x(x + y) + z(x + y) $

Now factor out common binomial:
$$
= (x + y)(x + z)
$$

✔ Correct.

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✔ Summary of All Answers:

| Part | Expression | Factored Form |
|------|-----------|----------------|
| (a) | $ 25a^2 - 16b^2 $ | $ (5a + 4b)(5a - 4b) $ |
| (b) | $ 9m^2 + 12mn + 4n^2 $ | $ (3m + 2n)^2 $ |
| (c) | $ 16p^2 - 24p + 9 $ | $ (4p - 3)^2 $ |
| (j) | $ (x+1)^2 - 9 $ | $ (x+4)(x-2) $ |
| (k) | $ (x-1)^2 - (x+1)^2 $ | $ -4x $ |
| (m) | $ x^2 + yz + xy + xz $ | $ (x+y)(x+z) $ |

All the given solutions are correct and well-explained using standard algebraic identities:

- Difference of squares: $ A^2 - B^2 = (A+B)(A-B) $
- Perfect square trinomials: $ A^2 + 2AB + B^2 = (A+B)^2 $
- Grouping method for four-term polynomials

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📌 Tips for Factorization:

1. Look for patterns like difference of squares or perfect squares.
2. Group terms when there are four or more terms.
3. Always check if the result can be simplified further.
4. Verify by expanding the factored form to see if you get back the original expression.

Let me know if you'd like to see any of these expanded to verify!