Factor the following expressions: (i) lx² + mx (iv) x² - ax - bx + ab

Question image

Understand the Problem

The question is asking to factor two algebraic expressions, specifically (i) lx² + mx and (iv) x² - ax - bx + ab. This involves simplifying the equations to identify their factors.

Answer

(i) \( x(lx + m) \), (iv) \( (x - a)(x - b) \)
Answer for screen readers

(i) The factored form of ( lx^2 + mx ) is ( x(lx + m) ).

(iv) The factored form of ( x^2 - ax - bx + ab ) is ( (x - a)(x - b) ).

Steps to Solve

  1. Factor the first expression ( lx^2 + mx )

Identify a common factor in the terms ( lx^2 ) and ( mx ). The variable ( x ) is present in both terms.

Extract ( x ) as a common factor:

$$ lx^2 + mx = x(lx + m) $$

  1. Factor the second expression ( x^2 - ax - bx + ab )

Rearrange the terms for easier factoring:

$$ x^2 - ax - bx + ab = x^2 + ab - ax - bx $$

Group the first two and the last two terms:

$$ = (x^2 - ax) + (-bx + ab) $$

  1. Factor each group

From the first group ( (x^2 - ax) ), factor out ( x ):

$$ x(x - a) $$

From the second group ( (-bx + ab) ), factor out ( -b ):

$$ -b(x - a) $$

  1. Combine the factored expressions

Now combine the two factored groups:

$$ x(x - a) - b(x - a) $$

Factor out the common binomial ( (x - a) ):

$$ = (x - a)(x - b) $$

(i) The factored form of ( lx^2 + mx ) is ( x(lx + m) ).

(iv) The factored form of ( x^2 - ax - bx + ab ) is ( (x - a)(x - b) ).

More Information

Factoring is an essential skill in algebra that simplifies expressions and helps solve equations. Recognizing common factors and grouping terms can significantly ease the factorization process.

Tips

  • Forgetting to look for common factors: Always check for a common factor before attempting to factor a polynomial.
  • Incorrectly grouping terms: Ensure terms are combined logically; rearranging them might help in identifying common factors.

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