Factor the following expressions: (i) lx² + mx (iv) x² - ax - bx + ab
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
- 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:
$$ lx^2 + mx = x(lx + m) $$
- 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:
$$ = (x^2 - ax) + (-bx + ab) $$
- 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 ):
$$ -b(x - a) $$
- 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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