Simplify x(3x^2+4)+2(7x-3)

The problem provided is an algebraic expression simplification exercise. You are being asked to combine like terms and perform the necessary distributive operations in order to simplify the expression into its most reduced form, following the standard rules of algebra. This typically involves expanding the parentheses by multiplying the terms outside the parentheses with those inside, then adding or subtracting the resulting terms as appropriate to reach the simplest version of the expression.

$x (3 x^{2} + 4) + 2 (7 x - 3)$

Answer

Expert–verified

Solution:

Step:1

Break down and simplify each component of the expression.

Step:1.1

Utilize the distributive property to expand: $x (3 x^{2}) + x \cdot 4 + 2 (7 x - 3)$

Step:1.2

Reorder using the commutative property: $3 x \cdot x^{2} + x \cdot 4 + 2 (7 x - 3)$

Step:1.3

Rearrange the multiplication order: $3 x \cdot x^{2} + 4 \cdot x + 2 (7 x - 3)$

Step:1.4

Combine like terms by adding exponents.

Step:1.4.1

Reposition $x^{2}$ : $3 (x^{2} x) + 4 \cdot x + 2 (7 x - 3)$

Step:1.4.2

Perform the multiplication of $x^{2}$ and $x$ .

Step:1.4.2.1

Exponentiate $x$ to the first power: $3 (x^{2} x^{1}) + 4 \cdot x + 2 (7 x - 3)$

Step:1.4.2.2

Apply the exponent rule $a^{m} a^{n} = a^{m + n}$ : $3 x^{2 + 1} + 4 \cdot x + 2 (7 x - 3)$

Step:1.4.3

Sum the exponents: $3 x^{3} + 4 \cdot x + 2 (7 x - 3)$

Step:1.5

Distribute the $2$ : $3 x^{3} + 4 x + 2 (7 x) + 2 \cdot (- 3)$

Step:1.6

Calculate the product of $7$ and $2$ : $3 x^{3} + 4 x + 14 x + 2 \cdot (- 3)$

Step:1.7

Calculate the product of $2$ and $- 3$ : $3 x^{3} + 4 x + 14 x - 6$

Step:2

Combine the terms $4 x$ and $14 x$ : $3 x^{3} + 18 x - 6$

Knowledge Notes:

The problem-solving process involves simplifying a polynomial expression. The key knowledge points and steps are:

Distributive Property: This property allows you to multiply a single term by each term within a parenthesis. For example, $a (b + c) = a b + a c$ .
Commutative Property of Multiplication: This property states that you can change the order of factors without changing the product. For example, $a b = b a$ .
Combining Like Terms: Terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients.
Exponent Rules: When multiplying like bases, you add the exponents. For example, $x^{m} \cdot x^{n} = x^{m + n}$ .
Simplification: This involves performing all possible operations, including distributing and combining like terms, to reduce the expression to its simplest form.

The steps taken in the solution follow these principles to simplify the given algebraic expression.