Multiplying Polynomials

Here you will find a concise collection of multiplying polynomials practice problems. Visit this page directly at hunkim.com/multiply

1. Expand and simplify the following polynomials:
   1. $-3x(x-1)$
      Solution
      $=-3x^2+3x$
   2. $2x^2(2-3x+4x^2)$
      Solution
      $=8x^4-6x^3+4x^2$
   3. $-(2-3x^2+2x^5-3x)$
      Solution
      $=-2x^5+3x^2+3x-2$
   4. $(x-3)(x-5)$
      Solution
      $=x^2-8x+15$
   5. $(3x-2)(x-3)$
      Solution
      $=3x^2-11x+6$
   6. $(2x-7)^2$
      Solution
      $=4x^2-28x+49$
   7. $2(3x-1)^2$
      Solution
      $=18x^2-12x+2$
   8. $2(3x-1)(x-2)$
      Solution
      $=6x^2-14x+4$
   9. $(x+2)(-2)(x-4)$
      Solution
      $=-2x^2+4x+16$
   10. $(x-1)(x^2+x+1)$
       Solution
       $=x^3-1$
   11. $(x^2-x-2)(x+1)$
       Solution
       $=x^3-3x-2$
   12. $(x^2+x+1)(1-x-x^2)$
       Solution
       $=-x^4-2x^3-x^2+1$
   13. $(x-1)^4$
       Solution
       $=x^4-4x^3+6x^2-4x+1$
2. Represent the product of $(2x-1)(x+2)$ using algebra tiles.
   Solution
   
   
3. The length of an edge of a cube is $x-1$
   1. Find the volume of the cube in expanded form
      Solution
      $=x^3-3x^2+3x-1$
   2. Find the area of the cube
      Solution
      $6x^2-12x+6$
4. Area of the shaded region below?
   
   Solution
   $=2x^2+14x+30$
5. See ramp below:
   
   1. Volume?
      Solution
      $=260x$
   2. Area including the bottom?
      Solution
      $c=\sqrt{116}=2\sqrt{29}$
      $A=\left(180+20\sqrt{29}\right) x+52$
6. The diameter of a circle is $2x+4$
   1. Area in expanded form?
      Solution
      $=\pi x^2+4\pi x+4\pi$
   2. Circumference?
      Solution
      $\pi(2x+4)=2\pi x+4\pi$
7. The area of the top of a cylinder is $9\pi$ m² and the height is $10x$. Find the lateral area of the cylinder.
   Solution
   $=60\pi x$ m²
8. The volume of a square-based pyramid is $3.\bar{3}$ m³.  Find the volume of a box that has the same height and base.
   Solution
   $10$
9. The volume of a cylinder is $120\pi x$ cm³.  Find the volume of a cone that has the same height and base.
   Solution
   $40\pi x$ cm³
10. Challenge:
    1. The diameter of earth is about $12.742x$ km.  Earth has a volume of about 1,083, 206, 916, 846 km³.
       1. Given the volume of a sphere is $V=\frac{4}{3}\pi r^3$, find $x$
          Solution
          $1000$
       2. Now estimate the surface area of earth given the area of a sphere is $A=4\pi r^2$
          Solution
          $\approx 510,064,479.9$ km²
    2. The surface area of the side of a cone is $A=\pi rs$. $s$ is the length of the diagonal slant of the cone.  Given the diameter of the cone’s base is 6 and the height of the cone is $4x$, find the entire area of the cone including the base.
       Solution
       $=3\pi \sqrt{16x^2+9}$
    3. A swimming pool can barely fit four unit circles on its base (each circle has a radius of 1 unit).  Given the depth of the pool is $x$, find the surface area of the pool liner that must cover the base and sides of the pool.  See top view of the pool below:
       
       Solution
       $2\pi x+\pi +12=(2x+1)\pi+12$

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