Simplify the Polynomial Expression: 5x³ + 3x²

Q: Why can't I just add the exponents 3 + 2 = 5?

Exponent addition only works when multiplying terms, like $ x^3 \cdot x^2 = x^5 $. Here we're adding terms, not multiplying them, so the exponents stay separate.

Q: What does 'expand' actually mean?

Expanding means writing out what the exponent tells us to do. The exponent 3 means multiply x by itself 3 times: $ x \cdot x \cdot x $.

Q: Is there a shorter way to write this?

The original form $ 5x^3 + 3x^2 $ is actually the shortest way! Expanding into multiplication shows the structure but makes it longer.

Q: Can I factor out anything from this expression?

Yes! You can factor out $ x^2 $: $ 5x^3 + 3x^2 = x^2(5x + 3) $. Both terms contain at least $ x^2 $ as a factor.

Q: What if x equals zero?

If $ x = 0 $, then both $ 5x^3 = 0 $ and $ 3x^2 = 0 $, so the entire expression equals 0.

Polynomial Expansion with Exponential Factors

Simplify the expression:

$5x^3 + 3x^2$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the expression:

$5x^3 + 3x^2$

Step-by-step solution

To simplify the expression $5x^3 + 3x^2$ , we can break it down into basic terms:

The term $5x^3$ can be written as $5 \cdot x \cdot x \cdot x$ .

The term $3x^2$ can be written as $3 \cdot x \cdot x$ .

Thus, the expression simplifies to $5 \cdot x \cdot x \cdot x + 3 \cdot x \cdot x$ .

Final Answer

$5\cdot x\cdot x\cdot x + 3\cdot x\cdot x$

Key Points to Remember

Essential concepts to master this topic

Exponent Rule: $x^3 = x \cdot x \cdot x$ and $x^2 = x \cdot x$
Technique: Expand each term: $5x^3 = 5 \cdot x \cdot x \cdot x$
Check: Count multiplication signs match original exponents: 3 x's for $x^3$ , 2 x's for $x^2$ ✓

Common Mistakes

Avoid these frequent errors

Combining unlike terms incorrectly
Don't add coefficients and exponents like 5 + 3 = 8 to get $8x^5$ ! This completely ignores exponent rules and creates an entirely different expression. Always expand each term separately: $x^3 = x \cdot x \cdot x$ and $x^2 = x \cdot x$ .

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

Why can't I just add the exponents 3 + 2 = 5?

Exponent addition only works when multiplying terms, like $x^3 \cdot x^2 = x^5$ . Here we're adding terms, not multiplying them, so the exponents stay separate.

What does 'expand' actually mean?

Expanding means writing out what the exponent tells us to do. The exponent 3 means multiply x by itself 3 times: $x \cdot x \cdot x$ .

Is there a shorter way to write this?

The original form $5x^3 + 3x^2$ is actually the shortest way! Expanding into multiplication shows the structure but makes it longer.

Can I factor out anything from this expression?

Yes! You can factor out $x^2$ : $5x^3 + 3x^2 = x^2(5x + 3)$ . Both terms contain at least $x^2$ as a factor.

What if x equals zero?

If $x = 0$ , then both $5x^3 = 0$ and $3x^2 = 0$ , so the entire expression equals 0.

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