ISEE Lower Level Math : Distributive Property

Study concepts, example questions & explanations for ISEE Lower Level Math

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Example Questions

Example Question #81 : Numbers And Operations

Multiply: \(\displaystyle (-x-3)(-x-7)\)

Possible Answers:

\(\displaystyle x^2+10x+21\)

\(\displaystyle -x^2-7x-21\)

\(\displaystyle x^2-10x-21\)

\(\displaystyle x^2-4x-21\)

\(\displaystyle x^2-4x+21\)

Correct answer:

\(\displaystyle x^2+10x+21\)

Explanation:

To multiply these terms use the Distributive property by applying the FOIL method. The FOIL method requires that we multiply each term and then combine like terms: \(\displaystyle (a+b)(c+d)=ac+ad+bc+bd\)

Note: FOIL stands for First, Outside, Inside, Last, where 

First\(\displaystyle =ac\)
Outside\(\displaystyle =ad\)
Inside\(\displaystyle =bc\)
Last\(\displaystyle =bd\)

\(\displaystyle ac=x^2,ad=7x,bc=3x,bd=21\)
Final solution equals: \(\displaystyle x^2+10x+21\)

Example Question #82 : Numbers And Operations

Evaluate: 

\(\displaystyle x(-x+y)\)

Possible Answers:

\(\displaystyle 2x+y\)

\(\displaystyle xy^2\)

\(\displaystyle x^2+y^2\)

\(\displaystyle -x^2+xy\)

\(\displaystyle 2xy\)

Correct answer:

\(\displaystyle -x^2+xy\)

Explanation:

To evaluate these terms apply the Distributive property: \(\displaystyle a(b+c)=ab+ac\)

Thus, the solution is: 

\(\displaystyle x(-x+y)=(x\times- x)+(x\times y)\)

\(\displaystyle x(-x+y)=-x^2+xy\)

Remember: we can not combine or add unlike variables or terms. For example, \(\displaystyle x+y\neq xy\)

Example Question #81 : Isee Lower Level (Grades 5 6) Mathematics Achievement

Expand the expression.

\(\displaystyle (x+2)(3x-5)\)

Possible Answers:

\(\displaystyle 3x^{2}-11x-10\)

\(\displaystyle 3x^2+x-10\)

\(\displaystyle 3x^{2}+x-3\)

\(\displaystyle 3x^{2}-x-10\)

\(\displaystyle 3x^{2}+11x+10\)

Correct answer:

\(\displaystyle 3x^2+x-10\)

Explanation:

Use FOIL (first, outer, inner, last) to expand.

First: \(\displaystyle (x)(3x)= 3x^{2}\)

Outside: \(\displaystyle (x)(-5) = -5x\)

Inside: \(\displaystyle (2)(3x)=6x\)

Last: \(\displaystyle (2)(-5) =-10\)

Sum the four terms into one expression.

\(\displaystyle 3x^{2}-5x+6x-10\)

Simplify by combining like terms.

\(\displaystyle 3x^2+x-10\)

Example Question #35 : Find Factor Pairs

Jack purchased \(\displaystyle 11\) tomato seeds and wants to make bags to sell at the local farmers’ market. How many different ways can Jack make seed bags with an equal number of seeds in each bag?

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle 2\)

\(\displaystyle 3\)

\(\displaystyle 11\)

\(\displaystyle 7\)

Correct answer:

\(\displaystyle 2\)

Explanation:

We will solve this problem by finding factor pairs. Factor pairs are composed of two numbers that are multiplied together to equal a product. List all the factor pairs of Jack’s seeds.

\(\displaystyle 1\times11=11\)

Do not forget to list their reciprocals.

\(\displaystyle 11\times1=11\)

Jack can make \(\displaystyle 2\) different seed bag combinations with an equal number of seeds in each bag.

Example Question #81 : Distributive Property

Jack purchased \(\displaystyle 18\) tomato seeds and wants to make bags to sell at the local farmers’ market. If he wants to have at least two bags to sell, many different ways can Jack make seed bags with an equal number of seeds in each bag?

Possible Answers:

\(\displaystyle 8\)

\(\displaystyle 5\)

\(\displaystyle 9\)

\(\displaystyle 7\)

\(\displaystyle 6\)

Correct answer:

\(\displaystyle 5\)

Explanation:

We will solve this problem by finding factor pairs. Factor pairs are composed of two numbers that are multiplied together to equal a product. List all the factor pairs of Jack’s seeds.

\(\displaystyle 1\times18=18\) -- but note that one bag of 18 seeds does not satisfy the condition that he wants to have at least two bags to sell, so this factorization does not count toward the answer.

\(\displaystyle 2\times9=18\) -- two bags of nine seeds each

\(\displaystyle 3\times6=18\) -- three bags of six seeds each

Do not forget to list their reciprocals.

\(\displaystyle 6\times3=18\) -- six bags of three seeds each

\(\displaystyle 9\times2=18\) -- nine bags of two seeds each

\(\displaystyle 18\times1=18\) -- eighteen bags of one seed each

Jack can make \(\displaystyle 5\) different seed bag combinations with an equal number of seeds in each bag.

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