Solution: Step 1: Extract the common factor Extract the factor of 2 from the expression x2 cdot 10x + 24. Step 1.1: Factor out 2 from x2 cdot 10x 2(x2 cdot 5x) + 24 Step 1.2: Factor out 2 from 24 2(x2 cdot 5x) + 2(12) Step 1.3: Combine the factored terms 2(x2 cdot 5x + 12) Step 2: Apply the commutative property 2(5x2 cdot x + 12) Step 3: Combine like terms by adding exponents Step 3.1: Rearrange the multiplication 2(5(x cdot x2) + 12) Step 3.2: Perform the multiplication of x and x2 Step 3.2.1: Represent x as x1 2(5(x1 cdot x2) + 12) Step 3.2.2: Apply the exponent rule am cdot an = am+n 2(5x1+2 + 12) Step 3.3: Add the exponents 2(5x3 + 12) Knowledge Notes: To solve the problem of factoring the polynomial x2 cdot 10x + 24, we apply several algebraic principles and properties: 1. Factorization: This is the process of breaking down an expression into a product of simpler factors. In this case, we factor out the greatest common factor, which is 2. 2. Distributive Property: This property allows us to factor out a common factor from terms in an expression. It states that a(b + c) = ab + ac. 3. Commutative Property of Multiplication: This property allows us to rearrange the factors in a multiplication without changing the product. It states that ab = ba. 4. Exponent Rules: Specifically, the power rule for exponents, which states that when multiplying two powers with the same base, you can add the exponents: am cdot an = am+n. 5. Combining Like Terms: This involves adding or subtracting terms with the same variable raised to the same power. By applying these principles …

Factor x^2*10x+24

The problem provided is a factoring problem, which requires breaking down the algebraic expression $ x^2 + 10x + 24 $ into a product of two binomials. Essentially, you are asked to find two binomial expressions that, when multiplied together, give you the original quadratic expression. This involves identifying two numbers that both add to give you the middle coefficient (in this case, 10) and multiply to give you the constant term (in this case, 24) in the quadratic expression.

$x^{2} \cdot 10 x + 24$

Answer

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Solution:

Step 1: Extract the common factor

Extract the factor of $2$ from the expression $x^{2} \cdot 10x + 24$.

Step 1.1: Factor out $2$ from $x^{2} \cdot 10x$

$2(x^{2} \cdot 5x) + 24$

Step 1.2: Factor out $2$ from $24$

$2(x^{2} \cdot 5x) + 2(12)$

Step 1.3: Combine the factored terms

$2(x^{2} \cdot 5x + 12)$

Step 2: Apply the commutative property

$2(5x^{2} \cdot x + 12)$

Step 3: Combine like terms by adding exponents

Step 3.1: Rearrange the multiplication

$2(5(x \cdot x^{2}) + 12)$

Step 3.2: Perform the multiplication of $x$ and $x^{2}$

Step 3.2.1: Represent $x$ as $x^{1}$

$2(5(x^{1} \cdot x^{2}) + 12)$

Step 3.2.2: Apply the exponent rule $a^{m} \cdot a^{n} = a^{m+n}$

$2(5x^{1+2} + 12)$

Step 3.3: Add the exponents

$2(5x^{3} + 12)$

Knowledge Notes:

To solve the problem of factoring the polynomial $x^{2} \cdot 10x + 24$, we apply several algebraic principles and properties:

Factorization: This is the process of breaking down an expression into a product of simpler factors. In this case, we factor out the greatest common factor, which is $2$.
Distributive Property: This property allows us to factor out a common factor from terms in an expression. It states that $a(b + c) = ab + ac$.
Commutative Property of Multiplication: This property allows us to rearrange the factors in a multiplication without changing the product. It states that $ab = ba$.
Exponent Rules: Specifically, the power rule for exponents, which states that when multiplying two powers with the same base, you can add the exponents: $a^{m} \cdot a^{n} = a^{m+n}$.
Combining Like Terms: This involves adding or subtracting terms with the same variable raised to the same power.

By applying these principles in a systematic way, we can factor the given polynomial expression.