Expand the Expression: 5x(x+2)(x+5) - Polynomial Multiplication

Q: Why don't I need to expand this expression?

Since the equation equals zero, you can use the zero product property directly! When a product of factors equals zero, at least one factor must be zero. This is much faster than expanding.

Q: What if the equation didn't equal zero?

If the right side wasn't zero, you'd need to move all terms to one side first to get zero on one side, then factor if possible. The zero product property only works when one side equals zero.

Q: How do I solve x + 2 = 0?

Subtract 2 from both sides: $ x + 2 - 2 = 0 - 2 $, so $ x = -2 $. Always isolate the variable by doing the opposite operation.

Q: Why are there three solutions instead of one?

This is a cubic equation (degree 3) because when expanded it would have $ x^3 $ as the highest power. Cubic equations can have up to 3 real solutions, and this one has exactly 3.

Q: Do I need to check all three answers?

Yes, always verify! Substitute each solution back into the original equation. If any factor becomes zero, the entire product becomes zero, confirming your answer is correct.

Zero Product Property with Factored Form

$5x(x+2)(x+5)=$

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Understand the problem

$5x(x+2)(x+5)=$

Step-by-step solution

Let's solve the given equation, noting that on the right side of the given equation is the number 0, and on the left side is a multiplication of algebraic expressions only:

$5x(x+2)(x+5)= 0$ From here we'll remember that the result of multiplication between expressions will yield 0 only if at least one of the multiplying expressions equals zero,

Therefore we'll get three simple equations and solve them by isolating the variable in each:

$\boxed{x=0}$ or:

$x+2=0\\ \boxed{x=-2}$

or:

$x+5=0\\ \boxed{x=-5}$

Therefore the correct answer is answer D.

Final Answer

All of the above

Key Points to Remember

Essential concepts to master this topic

Rule: If $ab = 0$ , then $a = 0$ or $b = 0$
Technique: Set each factor equal to zero: $5x = 0$ , $x+2 = 0$ , $x+5 = 0$
Check: Substitute each solution: $5(0)(2)(5) = 0$ ✓

Common Mistakes

Avoid these frequent errors

Expanding the expression instead of using zero product property
Don't multiply out $5x(x+2)(x+5)$ to get a cubic equation = unnecessary complexity! This creates a harder problem when the factored form already shows the solutions clearly. Always recognize when an equation equals zero and use the zero product property directly.

Practice Quiz

Test your knowledge with interactive questions

$$ 5x=0 $$

FAQ

Everything you need to know about this question

Why don't I need to expand this expression?

Since the equation equals zero, you can use the zero product property directly! When a product of factors equals zero, at least one factor must be zero. This is much faster than expanding.

What if the equation didn't equal zero?

If the right side wasn't zero, you'd need to move all terms to one side first to get zero on one side, then factor if possible. The zero product property only works when one side equals zero.

How do I solve x + 2 = 0?

Subtract 2 from both sides: $x + 2 - 2 = 0 - 2$ , so $x = -2$ . Always isolate the variable by doing the opposite operation.

Why are there three solutions instead of one?

This is a cubic equation (degree 3) because when expanded it would have $x^3$ as the highest power. Cubic equations can have up to 3 real solutions, and this one has exactly 3.

Do I need to check all three answers?

Yes, always verify! Substitute each solution back into the original equation. If any factor becomes zero, the entire product becomes zero, confirming your answer is correct.

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