Write in Standard Form (x-3)(x-4)(x+1)(x-2)
The question is asking you to take a polynomial expression that is currently factored into four binomial terms (x-3), (x-4), (x+1), and (x-2), and to expand these terms and their respective multiplications in order to write the expression in standard form. Standard form for a polynomial refers to a single polynomial equation with terms arranged in descending powers of x, starting with the highest power and moving to the lowest, with all like terms combined. The question requires knowledge of algebraic expansion and simplification.
$\left(\right. x - 3 \left.\right) \left(\right. x - 4 \left.\right) \left(\right. x + 1 \left.\right) \left(\right. x - 2 \left.\right)$
Answer
Solution:
Step 1:
To convert a polynomial into standard form, expand the expression and then order the terms from highest to lowest degree, in the form $a x^{n} + b x^{n-1} + \ldots + c$.
Step 2:
First, expand the binomials $(x - 3)(x - 4)$ using the FOIL (First, Outer, Inner, Last) method.
Step 2.1:
Distribute the terms: $(x(x - 4) - 3(x - 4))(x + 1)(x - 2)$.
Step 2.2:
Continue distribution: $(x \cdot x + x \cdot -4 - 3 \cdot x + 3 \cdot 4)(x + 1)(x - 2)$.
Step 2.3:
Keep distributing: $(x^2 - 4x - 3x + 12)(x + 1)(x - 2)$.
Step 3:
Combine like terms within the expanded expression.
Step 3.1:
Simplify each term within the parentheses.
Step 3.1.1:
Multiply $x$ by itself: $(x^2 - 4x - 3x + 12)(x + 1)(x - 2)$.
Step 3.1.2:
Rearrange the terms: $(x^2 - 7x + 12)(x + 1)(x - 2)$.
Step 3.1.3:
Combine like terms: $(x^2 - 7x + 12)(x + 1)(x - 2)$.
Step 4:
Expand the trinomial $(x^2 - 7x + 12)$ with the binomial $(x + 1)$ by multiplying every term of the first polynomial with every term of the second.
Step 5:
Simplify the resulting expression.
Step 5.1:
Simplify each term after the multiplication.
Step 5.1.1:
Multiply $x^2$ by $x$: $(x^3 + x^2 - 7x^2 - 7x + 12x + 12)(x - 2)$.
Step 5.1.2:
Multiply $x^2$ by $1$: $(x^3 + x^2 - 7x^2 - 7x + 12x + 12)(x - 2)$.
Step 5.1.3:
Multiply $x$ by $x$: $(x^3 + x^2 - 7x^2 - 7x + 12x + 12)(x - 2)$.
Step 5.1.4:
Multiply $-7$ by $1$: $(x^3 + x^2 - 7x^2 - 7x + 12x + 12)(x - 2)$.
Step 5.1.5:
Multiply $12$ by $1$: $(x^3 + x^2 - 7x^2 - 7x + 12x + 12)(x - 2)$.
Step 5.2:
Combine like terms: $(x^3 - 6x^2 + 5x + 12)(x - 2)$.
Step 6:
Expand the polynomial $(x^3 - 6x^2 + 5x + 12)$ with the binomial $(x - 2)$.
Step 7:
Simplify the resulting expression.
Step 7.1:
Simplify each term after the multiplication.
Step 7.1.1:
Multiply $x^3$ by $x$: $x^4 - 2x^3 - 6x^3 + 12x^2 + 5x^2 - 10x + 12x - 24$.
Step 7.2:
Combine like terms to get the final standard form: $x^4 - 8x^3 + 17x^2 + 2x - 24$.
Knowledge Notes:
Standard Form of a Polynomial: A polynomial is in standard form when its term degrees are in descending order, starting with the highest power.
FOIL Method: A technique to multiply two binomials. FOIL stands for First, Outer, Inner, Last, referring to the position of each term in the parentheses.
Distributive Property: A property that allows us to multiply a sum by multiplying each addend separately and then sum the products.
Combining Like Terms: The process of simplifying expressions by adding or subtracting terms that have the same variable raised to the same power.
Power Rule: In algebra, the power rule for exponents states that $a^{m} \cdot a^{n} = a^{m+n}$, which is used to simplify expressions with exponents.
Multiplication of Polynomials: Involves distributing each term of one polynomial to each term of the other and combining like terms.
