What is (x - a) (x - b) (x - c) equal to?
x 3- (a + b + c)x 2+ (bc + ca + ab)x - abc
Understanding the Expansion of \( (x - a)(x - b)(x - c) \)
The question asks us to find the expanded form of the algebraic expression \( (x - a)(x - b)(x - c) \). This is a product of three linear factors involving the variable \( x \) and constants \( a \), \( b \), and \( c \).
Step-by-Step Expansion Process
To expand this product, we can multiply the factors step by step. First, let's multiply the first two factors, \( (x - a)(x - b) \):
\[ (x - a)(x - b) = x(x - b) - a(x - b) \]
Applying the distributive property:
\[ = x \cdot x - x \cdot b - a \cdot x - a \cdot (-b) \]
\[ = x^2 - bx - ax + ab \]
Combining the terms with \( x \):
\[ = x^2 - (a + b)x + ab \]
Now, we multiply this result by the third factor, \( (x - c) \):
\[ (x^2 - (a + b)x + ab)(x - c) \]
Again, applying the distributive property, we multiply each term in the first parenthesis by \( x \) and then by \( -c \):
\[ = x(x^2 - (a + b)x + ab) - c(x^2 - (a + b)x + ab) \]
\[ = (x \cdot x^2 - x \cdot (a + b)x + x \cdot ab) - (c \cdot x^2 - c \cdot (a + b)x + c \cdot ab) \]
\[ = (x^3 - (a + b)x^2 + abx) - (cx^2 - c(a + b)x + cab) \]
Now, remove the parenthesis, being careful with the signs:
\[ = x^3 - (a + b)x^2 + abx - cx^2 + c(a + b)x - abc \]
\[ = x^3 - (a + b)x^2 - cx^2 + abx + (ac + bc)x - abc \]
Finally, group the terms with the same power of \( x \):
- Terms with \( x^3 \): \( x^3 \)
- Terms with \( x^2 \): \( -(a + b)x^2 - cx^2 = -(a + b + c)x^2 \)
- Terms with \( x \): \( abx + (ac + bc)x = (ab + ac + bc)x \)
- Constant term: \( -abc \)
Putting it all together, the expanded form is:
\[ x^3 - (a + b + c)x^2 + (ab + ac + bc)x - abc \]
We can write \( ac + bc \) as \( ca + bc \) or \( bc + ca \). So the term with \( x \) can be written as \( (ab + bc + ca)x \) or \( (bc + ca + ab)x \).
Matching with the Options
Let's compare our expanded form \( x^3 - (a + b + c)x^2 + (ab + bc + ca)x - abc \) with the given options:
- Option 1: \( x^3 - (a + b + c)x^2 + (bc + ca + ab)x - abc \)
- Option 2: \( x^3 + (a + b + c)x^2 + (bc + ca + ab)x + abc \)
- Option 3: \( x^3 - (bc + ca + ab)x^2 + (a + b + c)x - abc \)
- Option 4: \( x^3 + (bc + ca + ab)x^2 - (a + b + c)x - abc \)
Our result exactly matches Option 1.
Revision Table: Common Polynomial Expansions
| Expression | Expanded Form |
|---|---|
| \( (x+a)(x+b) \) | \( x^2 + (a+b)x + ab \) |
| \( (x-a)(x-b) \) | \( x^2 - (a+b)x + ab \) |
| \( (x+a)^2 \) | \( x^2 + 2ax + a^2 \) |
| \( (x-a)^2 \) | \( x^2 - 2ax + a^2 \) |
| \( (x-a)(x-b)(x-c) \) | \( x^3 - (a+b+c)x^2 + (ab+bc+ca)x - abc \) |
| \( (x+a)(x+b)(x+c) \) | \( x^3 + (a+b+c)x^2 + (ab+bc+ca)x + abc \) |
Additional Information: Roots and Coefficients
The expansion of \( (x - a)(x - b)(x - c) = x^3 - (a+b+c)x^2 + (ab+bc+ca)x - abc \) is a fundamental result. Notice the relationship between the roots (a, b, c) of the polynomial \( P(x) = (x-a)(x-b)(x-c) \) and its coefficients:
- The coefficient of \( x^2 \) is \( -(a+b+c) \), which is the negative of the sum of the roots.
- The coefficient of \( x \) is \( (ab+bc+ca) \), which is the sum of the products of the roots taken two at a time.
- The constant term is \( -abc \), which is the negative of the product of the roots.
This relationship is a specific case of Vieta's formulas, which connect the coefficients of a polynomial to sums and products of its roots. Understanding this relationship can help verify expansions or analyze polynomial properties.
Similar Questions
If (x 2- 1) is a factor of ax 4+ bx 3+ cx 2+ dx + e, then which one of the following is correct?
If α, β and γ are the zeros of the polynomial f(x) = ax 3+ bx 2+ cx + d, then α 2+ β 2+ γ 2is equal to
Let f(x) and g(x) be two polynomials (with real coefficients) having degree 3 and 4 respectively. What is the degree of f(x) g(x)
If α and β are the two zeros of the polynomial 25x 2– 15x + 2, then what is a quadratic polynomial whose zeros are (2α) -1 and (2β) -1 ?
The expression \(\frac{{\left( {{x^3} - 1} \right)\left( {{x^2} - 9x + 14} \right)}}{{\left( {{x^2} + x + 1} \right)\left( {{x^2} - 8x + 7} \right)}}\) simplifies to
The HCF and the LCM of two polynomials are 3x + 1 and 30x 3 + 7x 2 - 10x - 3 respectively. If one polynomial is 6x 2 + 5x + 1, then what is the other polynomial?
If (6x + 4y) / (6x - 4y) = 8/6 then what is the value of x 2 / y 2?
What is the sum of the linear factors (in x and y) of the expression
2x 2 + xy - 3y 2 ?
What should be added to \(\frac{1}{{(x - 2)(x - 4)}}\) to get \(\frac{2x - 5}{{(x^2-5x+6)(x - 4)}}\) ?
For what value of k can the expression x 3+ kx 2– 7x + 6 be resolved into three linear factors?
Important Questions from Polynomials
If y 2= y + 7, then what is the value of y 3?
Factorize x 2- y 2- 9z 2+ 6yz
If one of the zeros of the polynomial x 3+ ax 2+ bx + c is - 1, then the product of other two zeros is equal to :
If a(a + b + c) 2 = 1792; b(a + b + c) 2 = 1536; c(a + b + c) 2 = 768, then what will be the value of b?
If x = 3 so, what is the value of x 2 + 2x + 5 ?
