Symplifying Algebraic Expressions: Difficult Problems with Solutions

Solution:
|-2| = 2
[tex]2 - (-3) = 2 + 3 = 5[/tex]

Solution:
We rearrage the terms and group in order to get [tex](5x^2-3x^2)+(16x-8x)+(7+1)=2x^2+8x+8=2(x^2+4x+4) = 2(x+2)^2=2\cdot(-5+2)^2=2\cdot(3^2)=2\cdot9=18[/tex]

Solution:
We regroup and combine: [tex]x^2+5x+8+3x^2+10x+1=(x^2+3x^2) + (5x+10x)+(8+1)=4x^2+15x+9=4\cdot(-4)^2+15\cdot(-4)+9=4\cdot 16-15\cdot 4+9=64-60+9=13[/tex]

Solution:
We rearrange and group: [tex](5y^2-3y^2)+(-8y+3y)+3=2y^2-5y+3=2\cdot (3^2)-5\cdot 3+3=18-15+3=6[/tex]

Solution:
We regroup: [tex](-5a^2b+8a^2b)+(8ab^2-15ab^2)+(2-1)=3a^2b-7ab^2+1[/tex]. Now we parenthesize, since [tex]ab[/tex] is a common factor of [tex]a^2b[/tex] and [tex]ab^2[/tex]: [tex]3ab \times a-7ab \times b+1=ab(3a-7b)+1=10 \times 2+1=21[/tex]

Solution:
First, we present [tex]-(x^2-4x+1)[/tex] as [tex]+(-1) \times (x^2-4x+1)=-x^2+4x-1[/tex]. Then we add this to [tex]3x^2-5x+2[/tex] to get: [tex]3x^2-5x+2-x^2+4x-1=2x^2-x+1=2 \times (5^2)-5+1=2 \times 25-4=50-4=46[/tex]

Solution:
We regroup: [tex](x^3-x^3)+(-5x^2+4x^2)+(8xz-8xz)+(z^2-z^2)+2=-x^2+2=-(3^2)+2=-9+2=-7[/tex].

Solution:
We'll first rewrite each of the terms:
[tex](z-2)(z+2)=z^2-4.[/tex]
[tex](3z-5)(2z+1)=3z(2z+1)-5(2z+1)=6z^2+3z-10z-5=6z^2-7z-5[/tex]. Now we add them together:
[tex]z^2-4+6z^2-7z-5-z^2+5z+1=(z^2+6z^2-z^2)+(-7z+5z)+(-4-5+1)=6z^2-2z-8=6 \times (10^2)-2 \times 10-8=600-20-8=572[/tex]

Solution:
Directly plugging in the values for [tex]x[/tex] and [tex]y[/tex] gives us: [tex](3\cdot 2+3)(2\cdot 2+1)-(3\cdot 3+2)(6\cdot 2+3)=(6+3)(4+1)-(9+2)(12+3)=9\cdot 5-11\cdot 15=5\cdot (9-11\cdot 3)=-5\cdot (33-9)=-5\cdot 24=-120[/tex]

Solution:
[tex](3m - 2)(2m + 5) = 3m(2m + 5) - 2(2m + 5) = 6m^{2} + 15m-4m-10 = 6m^{2}+11m-10[/tex]

Solution:
[tex](3x - 5)(2x^{2} + 7x - 5) = 3x(2x^{2})+3x(7x)+3x(-5)+(-5)(2x^{2})+(-5)(7x)+ (-5)(-5) = 6x^{3} + 21x^{2} - 15x - 10x^{2} - 35x + 25 = 6x^{3} + 11x^{2} - 50x + 25[/tex]

Solution:
[tex](3p -2y)(2p + 3y) = (3p)(2p)+(3p)(3y)+(-2y)(2p)+(-2y)(3y) = 6p^{2} + 9py - 4py - 6y^{2} = 6p^{2} + 5py - 6y^{2}[/tex]

Solution:
[tex](9p - 7q)(5p - 4q) = 45p^{2} - 36pq - 35pq + 28q^{2} = 45p^{2} - 71pq + 28q^{2}[/tex]

Solution:
[tex](2m - 3n)^{2} = (2m)^{2} - 2(2m)(3n) + (3n)^{2} = 4m^{2} - 12mn + 9n^{2}[/tex]

Solution:
[tex]\frac{10x^4mn^2}{5x^3mn} = \frac{10}{5} \times \frac{x^4}{x^3} \times \frac{m}{m} \times \frac{n^2}{n} = 2\cdot x\cdot 1\cdot n = 2xn[/tex]

Solution:
[tex]\frac{9xy - 6x^2}{3x} = \frac{9xy}{3x} - \frac{6x^2}{3x} = 3y - 2x[/tex]

Solution:
[tex](2x+5)^2=4x^2+20x+25[/tex]
[tex]4(x+1)(x+3)=4(x^2+4x+3)=4x^2+16x+12[/tex]
[tex]4\left(x+2\frac{1}{2}\right)=4x+10[/tex]
[tex]4x^2+20x+25-4x^2-16x-12-4x-10=3[/tex]
The result does not contain [tex]x[/tex], so the value is always [tex]3[/tex].

Solution:
[tex]8x(x-1)(x+1)=8x(x^2-1)=8x^3-8x[/tex]
[tex](2x-1)^3=8x^3-12x^2+6x-1[/tex]
[tex]A=8x^3-8x-8x^3+12x^2-6x+1-12x^2=-14x+1[/tex]

Solution:
[tex]a^3-b^3=(a-b)(a^2+ab+b^2)[/tex], [tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]
Group the factors suitably:
[tex][(x-1)(x^2+x+1)]\times [(x+1)(x^2-x+1)]=(x^3-1)(x^3+1)=x^6-1[/tex]

Solution:
[tex](x-2z)(x^2+2xz+4z^2)=x^3-8z^3[/tex]
[tex](x-2z)^3=x^3-6x^2z+12xz^2-8z^3[/tex]
[tex]x^3-8z^3-x^3+6x^2z-12xz^2+8z^3+12xz^2=6x^2z[/tex]

Solution:
First we simplify:
[tex](x-2)^3=x^3-6x^2+12x-8[/tex]
[tex](x-4)(x+1)(x-1)=(x-4)(x^2-1)=x^3-4x^2-x+4[/tex]
[tex]A=x^3-6x^2+12x-8-x^3+4x^2+x-4+2x^2=13x-12[/tex]
[tex]x=-\frac{12}{13}[/tex]: [tex]A=13\times \left(-\frac{12}{13}\right)-12=-12-12=-24[/tex]

Solution:
[tex](1+x^2)(x^2-1)=x^4-1[/tex]
[tex](1-x^2)^2=1-2x^2+x^4[/tex]
[tex](3x^5-x^3)\div (-x)^3=(3x^5-x^3)\div (-x^3)=-3x^2+1[/tex]
[tex]x^4-1-(1-2x^2+x^4)-(-3x^2+1)=x^4-1-1+2x^2-x^4+3x^2-1=5x^2-3[/tex]
The standard form [tex]5x^2-3[/tex] has [tex]2[/tex] monomials.

Solution:
Instead of expanding, notice that the expression has the form [tex]a^2-2ab+b^2=(a-b)^2[/tex] with [tex]a=2x+3[/tex] and [tex]b=2x-5[/tex]:
[tex]\big((2x+3)-(2x-5)\big)^2=8^2=64[/tex]
The value is [tex]64[/tex] for every [tex]x[/tex].

Solution:
[tex](x-1)^3=x^3-3x^2+3x-1[/tex]
[tex](x^2+2x+4)(x-2)=x^3-8[/tex]
[tex]-3m(1-x)=-3(x+1)(1-x)=-3(1-x^2)=3x^2-3[/tex]
[tex]P=x^3-3x^2+3x-1-x^3+8-3x+3x^2-3=4[/tex]
[tex]P=4[/tex] for every [tex]x[/tex], so the value does not depend on [tex]x[/tex].

Solution:
[tex]x^2(x^2+32)-x^2(x+3)+x(7x^3+x^2)+16=x^4+32x^2-x^3-3x^2+7x^4+x^3+16=8x^4+29x^2+16[/tex]
[tex]x^4\ge 0[/tex] and [tex]x^2\ge 0[/tex], so [tex]8x^4+29x^2+16\ge 16>0[/tex]. The values are positive for every [tex]x[/tex] (and the polynomial is of fourth degree).

Solution:
[tex]A=3y^4-3y^2-2y-1-3y^4+5y^2+2y=2y^2-1[/tex]
Since [tex]y^2\ge 0[/tex], we have [tex]2y^2-1\ge -1[/tex], with equality for [tex]y=0[/tex].
Answer: [tex]-1[/tex]

Solution:
Terms with [tex]x^2[/tex] come only from the products:
[tex]x^2\times 2+ax\times (-5x)+(-3)\times x^2=(2-5a-3)x^2=(-1-5a)x^2[/tex]
[tex]-1-5a=0 \Rightarrow a=-\frac{1}{5}[/tex]